\( \newcommand\R{\mathbb{R}} \newcommand\Q{\mathbb{Q}} \newcommand\Z{\mathbb{Z}} \newcommand\N{\mathbb{N}} \newcommand\ve[1]{\boldsymbol{#1}} \newcommand\norm[1]{\left\Vert #1 \right\Vert} \def\u{\ve{u}} \def\v{\ve{v}} \def\w{\ve{w}} \def\C{\mathcal{C}} \def\diam{\operatorname{diam}} \)
A sequence is a function whose domain is the set of natural numbers. The values of the function are called terms of the sequence. \[ \alpha\colon\N \to S,\quad n \mapsto \alpha_n.\] We denote \(\alpha_n = \alpha(n)\), and denote the whole sequence as \(\{\alpha_n\}\) or \((\alpha_n)_{n \in \N}\).
If the image of \(\alpha\) is bounded below or above, we say that the sequence is bounded below or above. If the sequence is bounded both below and above, then we say that the sequence is bounded.
A sequence \(\{\alpha_n\}\) in a metric space \((M, d)\) is said to converge to a point \(p \in M\), written \(\alpha_n \to p\), if for every \(r > 0\), there exists \(N \in \N\) such that for all \(n \geq N\), we have \(\alpha_n \in B_r(p)\).
Note that a sequence converges to a point \(p\) only if \(p\) is a limit point of the set \(\{\alpha_1, \alpha_2, \dots\}\). We write \[ \lim_{n \to \infty} \alpha_n = p. \]
Also note that if \(\alpha_n \to p\) in \(M\), then the sequence \(d(\alpha_n, p) \to 0\) in \(\R\).
A divergent sequence is one which does not converge.
If \(\alpha_n \to a\) and \(\alpha_n \to b\) in a metric space \((M, d)\), then \(a = b\).
If \(\alpha_n\) converges, then \(\alpha_n\) is bounded.
If \(\alpha_n \to a\) and \(\beta_n \to b\), then \(\alpha_n + \beta_n \to a + b\) and \(\alpha_n\beta_n \to ab\). Additionally, if \(\beta_n \neq 0\) and \(b \neq 0\), then \(\alpha_n/\beta_n \to a /b\).
If \(\ell\) is a limit point of \(S\), then there is a sequence \(\{s_n\}\) in \(S\) such that \(s_n \to \ell\).
Given a sequence \(\{\alpha_n\}\), consider an increasing sequence \(n_k\) of positive integers. Then, the sequence \(\{\alpha_{n_k}\}\) is called a subsequence of \(\{\alpha_n\}\). If \(\{\alpha_{n_k}\}\) converges, its limit is called the sub-sequential limit of \(\{\alpha_n\}\).
Let \((M, d)\) be a compact metric space and let \(\{\alpha_n\}\) be a sequence in \(M\). Then, some subsequence of \(\{\alpha_n\}\) converges to a point of \(M\).
Every bounded sequence in \(\R^n\) has a convergent subsequence.
Let \((M, d)\) be a metric space and let \(\{\alpha_n\}\) be a convergent sequence in \(M\). Then for every \(\epsilon > 0\), there exists \(N \in \N\) such that \(d(\alpha_m, \alpha_n) < \epsilon\) if \(\min(m, n) \geq N\).
A sequence \(\{\alpha_n\}\) in a metric space \((M, d)\) is called a Cauchy sequence if for every \(\epsilon > 0\), there is an integer \(N \in \N\) such that \(d(\alpha_m, \alpha_n) < \epsilon\) if \(\min(m, n) \geq N\).
If a subsequence \(\{\alpha_{n_k}\}\) of a Cauchy sequence \(\{\alpha_n\}\) converges to a limit point \(\ell\), then the entire sequence \(\{\alpha_n\}\) converges to \(\ell\).
A metric space \((M, d)\) is called Cauchy complete if every Cauchy sequence in \(M\) converges in \(M\).
Every compact subset of a metric space is Cauchy complete.
All Euclidean spaces are Cauchy complete.
A sequence in a Euclidean space \(\R^n\) converges iff it is Cauchy.
Let \(\C\) denote the set of all rational Cauchy sequences. Let \(\{\alpha_n\}, \{\beta_n\} \in \C\). We say that the sequences \(\{\alpha_n\} \sim \{\beta_n\}\) if \(|\alpha_n - \beta_n| \to 0\). It is easily verified that \(\sim\) is an equivalence relation.
We define \(\R\) as the quotient space of this relation, i.e. \[ \R = \C /\sim. \]
The rationals are embedded in the usual way, and addition and multiplication are defined termwise.
The order on \(\R\) is defined as follows: \(\{\alpha_n\} > \{\beta_n\}\) if there exists \(N\) such that \(\alpha_n > \beta_n\) for all \(n \geq N\), and \(|\alpha_n - \beta_n| \not\to 0\).
Let \(S\) be a nonempty subset of \(\Q\) and let \(u_1\in \Q\) be an upper bound of \(S\). Let \(s\in S\) and let \(\ell_1 \in \Q\) be such that \(\ell_1 < s\). We define two rational Cauchy sequences \(\{u_n\}\) and \(\{\ell_n\}\) as follows \[ \begin{align} u_{n + 1} &= \begin{cases} (u_n + \ell_n)/2, &\text{if this is an upper bound of }S, \\ u_n, &\text{otherwise} \end{cases},\\ \ell_{n + 1} &= \begin{cases} (u_n + \ell_n)/2, &\text{if this is not an upper bound of }S, \\ \ell_n, &\text{otherwise} \end{cases}. \end{align}\] It follows that all \(u_n\) are upper bounds of \(S\), and none of \(\ell_n\) are.
Suppose \(\{u_n\} < \{s_n\}\) for some \(\{s_n\} \in S\). This would mean that for some sufficiently large \(n_0\), \(u_{n_0} < s_{n_0}\), which is a contradiction. Thus, \(\{u_n\}\) is an upper bound of \(S\).
Also note that \[ u_n - \ell_n = \frac{u_1 - \ell_1}{2^{n - 1}}, \] so \(\{u_n\} \sim \{\ell_n\}\). Thus, \(\{u_n\} = \{\ell_n\}\) in \(\R\).
Let \(\{a_n\} \in \R\) be an upper bound of \(S\) such that \(\{a_n\} < \{u_n\}\). It follows that \(\{a_n\} < \{\ell_n\}\) in \(\R\). This implies that there exists \(\epsilon > 0\) and \(N_1 \in \N\) such that \[ \ell_n - a_n > \epsilon, \] for all \(n \geq N_1\). Since \(\{\ell_n\}\) is a Cauchy sequence, there exists \(N_2 \in \N\) such that for all \(m, n \geq N_2\), \(|\ell_m - \ell_n| < \epsilon/2\). Thus, setting \(N = \max(N_1, N_2)\), we see that for all \(n \geq N\), we have \[ \ell_n - \ell_N < \epsilon/2. \] Therefore, \[\ell_n - a_n \geq (\ell_N - \ell_n) - (\ell_n - a_n) > \epsilon/2, \] for all \(n \geq N\). This means that \(\{a_n\}\) is greater than the constant sequence \(\{\ell_N\}\). However, \(\ell_N\) was not an upper bound of \(S\), so neither can \(\{a_n\}\) be an upper bound of \(S\), which is a contradiction. Thus, \(\{u_n\}\) is the least upper bound of \(S\). This shows that \(\R\) has the least upper bound property, i.e. \(\R\) is complete.
A subset \(S\) of a metric space \((M, d)\) is called sequentially compact if every sequence of points in \(S\) has a convergent subsequence which converges to a point in \(S\).
Note that the Heine-Borel theorem guarantees that in Euclidean spaces, compactness and sequential compactness are equivalent. This is indeed true for all metric spaces.
Let \(S \subseteq M\) be nonempty, and let \(D = \{d(x, y): x, y \in S\}\). We define the dimeter of \(S\) as \(\sup{D}\).
Let \(\{\alpha_n\}\) be a sequence in a metric space \((M, d)\) and for \(N \in \N\), let \(S_N = \{\alpha_n: n \geq N\}\). Then, \(\{\alpha_n\}\) is a Cauchy sequence iff \[ \lim_{N \to \infty} \diam{S_N} = 0. \]
Let \(S\) be a subset of a metric space \((M, d)\). Then, \(\diam{\overline{S}} = \diam{S}\).
If \(K_n\) is a nested sequence of compact sets in \(M\) such that \(\lim_{n \to \infty} \diam{K_n} = 0\), then there exists a point \(p \in M\) such that \[ \bigcap_{n \in \N} K_n = \{p\}. \]
Every nondecreasing or nonincreasing sequence sequence is called monotone. Every bounded monotonic sequence converges in \(\R\).
Let \(S\) be the set of all subsequential limits of the sequence \(\{\alpha_n\}\) in a metric space \((M, d)\). Then, \(S\) is closed.
Let \(\{\alpha_n\}\) and \(\{\beta_n\}\) be real sequences such that for all \(M \in \R\), there exists \(N \in \N\) such that for every \(n \geq N\), \(\alpha_n \geq M\) and \(\beta_n \leq M\). We say that \(\{\alpha_n\}\) diverges to \(\infty\) and \(\{\beta_n\}\) diverges to \(-\infty\). We write \[ \alpha_n \to \infty, \qquad \beta_n \to -\infty. \]
Again, let \(S\) denote the set of all subsequential limit of \(\{\alpha_n\}\). If any subsequence of diverges to \(\pm \infty\), we let \(\pm\infty \in S\). Thus, \(s^* = \sup{S}\) and \(s_* = \inf{S}\) exist in the extended real number system. These are called the upper and lower limits of the sequence \(\{\alpha_n\}\). These are denoted as \[ \limsup_{n \to \infty} \alpha_n = s^*, \qquad \liminf_{n \to \infty} \alpha_n = s_*. \]
In general, \(\alpha_n \to \ell\) iff \(s^* = s_* = \ell\).
If \(s^* \neq s_*\), we say that the sequence \(\{\alpha_n\}\) oscillates. The oscillation \(\omega(\{\alpha_n\})\) of the sequence \(\{\alpha_n\}\) is defined as \(s^* - s_*\) if both are finite, else \(\infty\).
With \(S\), \(s^*\) and \(s_*\) defined as before, then \(s^* \in S\). If \(x > s^*\), then there exists an integer \(N\) such that for all \(n \geq N\), we have \(\alpha_n < x\). There is no other element of the extended reals which satisfies both of these properties.
Given a sequence \(\{\alpha_n\}\) in \(\R\), we define the \(k^\text{th}\)-partial sum as \(s_k = \sum_{n = 1}^k \alpha_n\).
If the sequence of the partial sums \(\{s_k\}\) converges, i.e. \(s_k \to s\), we say that the series \(\alpha_1 + \alpha_2 + \dots \) converges to \(s\). Otherwise, we say that it diverges.
The Cauchy criterion for the sequence \(\{s_k\}\) implies that the series \(\sum_{n = 1}^\infty \alpha_n\) converges iff for every \(\epsilon > 0\), there exists an integer \(N\) such that for all \(n > m \geq N\), we have \[ \left|\sum_{j = m + 1}^n \alpha_j \right| < \epsilon. \]
In particular, if the series \(\sum_{n = 1}^\infty \alpha_n\) converges, then \(\alpha_n \to 0\).
The Monotone Convergence theorem implies that a series of nonnegative reals converges iff its partial sums forms a bounded sequence.
Consider the geometric series \(\sum_{n = 1}^\infty a^n\). Note that the partial sums are \[s_k = \sum_{n = 1}^k a^n = \frac{1 - a^{k + 1}}{1 - a}.\] When \(a \neq 0\), \(-1 < a < 1\), the series converges to \(\ell = 1 /(1 - a)\). Note that \[ |s_k - \ell| = |\ell| |a^{k + 1}|. \] For \(a > 1\) and \(a < 1\), note that \(a^n \not\to 0\), hence the series diverges.
The comparison test states that for two real sequences \(\{\alpha_n\}\) and \(\{\beta_n\}\), suppose that \(|\alpha_n| < \beta_n\) for all \(n \geq N \in \N\). Then, if the series \(\sum \beta_n\) converges, so does \(\sum \alpha_n\). Also, if \(0 \leq \alpha_n < \beta_n\) for all \(n \geq N \in \N\) and \(\sum \alpha_n\) diverges, so does \(\sum \beta_n\).
The Cauchy condensation test states that if \(\{\alpha_n\}\) is a nonincreasing sequence of non-negative reals, the series \(\sum \alpha_n\) converges iff the series \(\sum 2^n\alpha_{2^n}\) converges.
The series \(\sum 1/n^j\) converges iff \(j > 1\). This can be shown using the Cauchy condensation test, whereby the given series converges iff \(\sum 2^n /(2^n)^j = \sum 2^{(1 - j)n}\) converges. This is a geometric series, which converges iff \(2^{1 - j} < 1\), i.e. \(j > 1\). Note that we can only use the Cauchy condensation test when \(j > 0\), but the series diverges trivially when \(j \leq 0\) since the terms do not go to zero.
The root test states that for a real sequence \(\{\alpha_n\}\), the series \(\sum \alpha_n\) converges if \(\limsup_{n \to \infty} |\alpha_n|^{1/n} < 1\) and diverges if \(\limsup_{n \to \infty} |\alpha_n|^{1/n} > 1\). No conclusion can be made if the limit is exactly \(1\).
The D'Alembert ratio test states that for a real sequence \(\{\alpha_n\}\) such that \(\alpha_n \neq 0\) for \(n \geq N\), the series converges if \(\limsup_{ n\to \infty} |\alpha_{n + 1} / \alpha_n| < 1\) and diverges if \(\limsup_{ n\to \infty} |\alpha_{n + 1} / \alpha_n| > 1\). No conclusion can be made if the limit is exactly \(1\).
For a real sequence \(\{\alpha_n\}_{n = 0}^\infty\), the series \[ \sum_{n = 0}^\infty \alpha_n x^n\] is called the power series of \(\{\alpha_n\}\).
Suppose \(a = \limsup_{n \to \infty} |\alpha_n|^{1/n}\) for such a sequence. Let \(R = 0\) if \(a = \infty\), \(R = \infty\) if \(a = 0\), and \(R = 1/a\) otherwise. The Cauchy Hadamard theorem states that the power series \(\sum \alpha_nx^n\) converges if \(|x| < R\) and diverges if \(|x| > R\). The quantity \(R\) is called the radius of convergence of the power series.
The exponential function is defined as \[ \exp(x) = \sum_{n = 0}^\infty \frac{x^n}{n!}. \] Note that the ratio test gives that as \(n \to \infty\), \[ \left|\frac{x^{n + 1}}{(n + 1)!}\cdot\frac{n!}{x^n}\right| = \left|\frac{x}{n + 1}\right| \to 0. \] Thus, the series converges everywhere, i.e. has a radius of convergence \(R = \infty\).
The number \(e = \exp(0)\) is called Euler's number.
Let \(\{\alpha_n\}\) be a real sequence. We say that the series \(\sum \alpha_n\) converges absolutely if the series \(\sum |\alpha_n|\) converges.
If a real series converges absolutely, it also converges.
A series \(\sum \alpha_n\) converges conditionally if it converges, but the series \(\sum |\alpha_n|\) diverges.
Abel's Lemma states that given two sequences \(\{\alpha_n\}_{n = 0}^\infty\) and \(\{\beta_n\}_{n = 0}^\infty\), define the \(k^\text{th}\) partial sums \[ s_k = \sum_{n = 0}^k \alpha_n. \] Also define \(s_k = 0\) if \(k \neq \N\). Then, for \(0 \leq m \leq n\), we have \[ \sum_{j = m}^n \alpha_j\beta_j = \sum_{j = m}^{n - 1} s_j(\beta_j - \beta_{j + 1}) + s_n\beta_n - s_{m - 1}\beta_m. \]
Suppose that \(\{s_k\}\) is bounded, the sequence \(\{\beta_n\}\) is nonincreasing, and \(\beta_n \to 0\). Then, \(\sum \alpha_j\beta_j\) converges.
Let \(\{\alpha_n\}\) be a nonincreasing sequence where \(\alpha_n \to 0\). Let the radius of convergence of the power series \(\sum\alpha_nx^n\) be \(1\). Then, this power series also converges at \(x = -1\).
We see that the alternating harmonic series \(\sum (-1)^n /n\) converges.
Let \(\{\alpha_n\}\) be a real sequence such that the sequence \(\{|\alpha_n|\}\) is nonincreasing, \(a_{2m - 1} > 0\) and \(a_{2m} < 0\) for all \(m \in \N\), and \(a_n \to 0\), then the series \(\sum \alpha_n\) converges.
Let \(f\colon\N\to\N\) be a bijection. We say that the series \(\sum \alpha_{f(n)}\) is a rearrangement of the series \(\sum \alpha_n\).
Let \(\sum \alpha_n\) be an absolutely convergent series, and let \(f\colon\N\to\N\) be a bijection. Then, the rearranged series \(\sum\alpha_{f(n)}\) is also absolutely convergent, and converges to the same value as the original series.
Let \(\sum \alpha_n\) be conditionally convergent. Then for each pair of \(a\) and \(b\) in the extended reals with \(a \leq b\), the given series has a rearrangement \(\sum \alpha_{f(n)}\) with partial sums \(s_k'\) such that \(\liminf_{k \to \infty} s_k' = a\), and \(\limsup_{k \to \infty} s_k' = b\).
Let \((M, d)\) and \((M', d')\) be metric spaces. Let \(S \subseteq M\), \(S' \subseteq M'\) and let there be a function \(f\colon S \to S'\). Let \(a\) be a limit point of \(S\). For \(b \in S'\), we write \(f(x) \to b\) as \(x \to a\) or \(\lim_{x \to a}f(x) = b\) if the preimage of every open ball around \(b\) contains an open ball around \(a\).
Note that \(b\) may not be an element of \(S'\). Neither does \(a\) have to be an element of \(S\).
Equivalently, we say that \(\lim_{x \to a}f(x) = b\) iff for every sequence \(\{\alpha_n\}\) in \(S\) such that \(\lim_{n \to \infty} \alpha_n = a\) and \(\alpha_n \neq a\) for all but finitely many \(n\), we have \(\lim_{n \to \infty} f(\alpha_n) = b\).
The function \(f\colon S \to S'\) is said to be continuous at \(a \in S\) if for every open set \(\mathcal{O}' \subseteq S'\) containing \(f(a)\), there exists an open set \(\mathcal{O} \subseteq S\) containing \(a\) such that \(\mathcal{O} \subseteq f^{-1}(\mathcal{O}')\).
If \(f\) is continuous at every point of \(S\), we say that \(f\) is continuous on \(S\).
Thus, for \(f\colon M \to M'\), \(f\) is continuous on \(M\) iff the inverse images of all open sets are open.
This immediately means that \(f\) is continuous iff the inverse images of all closed sets are closed.
Let \(a, b \in \R\) with \(a < b\). Let \(f\colon (a, b) \to M\) be a function. For \(x \in [a, b)\), we write \(f(x+) = \alpha\) if \(f(\alpha_n) \to \alpha\) for all sequences \(\{\alpha_n\}\) in \((x, b)\) such that \(\alpha_n \to x\). This is called the right hand limit.
For \(f\colon (a, b) \to M\) suppose \(f\) is discontinuous at \(x\). If both \(f(x+)\) and \(f(x-)\) exist, then \(f\) has a discontinuity of the first kind, a simple discontinuity at \(x\). Otherwise, we say that \(f\) has a discontinuity of the second kind at \(x\).
A connected space is one which cannot be expressed as the union of two or more disjoint, non-empty open sets. More precisely, a disconnected space is the union of two disjoint non-empty open sets.
The continuous image of a connected set is connected.
Every non-empty interval in \(\R\) is connected. In addition, every connected subset of \(\R\) is an interval.
Let \(a, b \in \R\) with \(a < b\) and let \(f\colon [a, b] \to \R\) be continuous. Let \(\alpha = \min(f(a), f(b))\) and \(\beta = \min(f(a), f(b))\). Then, for every \(y \in [\alpha, \beta]\), there exists \(x \in [a, b]\) such that \(f(x) = y\).
The continuous image of a compact set is compact.
A function \(f\colon S \to \R^n\) is called bounded if there exists \(M > 0\) such that \(\norm{f(x)} \leq M\) for all \(x \in S\).
Every continuous function from a compact set to a Euclidean space is bounded.
Let \((M, d)\) be a compact metric space and let \(f\colon M \to \R\) be continuous. Then \(f\) attains both its minimum and its maximum on \(M\).
Let \((M, d)\) be compact and let \(f\colon M \to M'\) be a continuous bijection. Then, \(f^{-1}\colon M' \to M\) defined by \(f^{-1}(f(x)) = x\) for all \(x \in M\) is also continuous.
Let \((M, d)\) and \((M', d')\) be metric spaces and \(f\colon M \to M'\) be a function. We say that \(f\) is uniformly continuous if for every \(\epsilon > 0\), there exists a \(\delta > 0\) such that \[ d'(f(x), f(y)) < \epsilon \text{ for all } x, y \in M \text{ with } d(x, y) < \delta. \]
If \(f\colon M \to M'\) is not uniformly continuous, then there are sequences \(\{x_n\}\) and \(\{y_n\}\) in \(M\) and some \(\epsilon > 0\) such that \(d(x_n, y_n) \to 0\) but \(d'(f(x_n), f(y_n)) \geq \epsilon\) for all \(n \in \N\).
The Mean Value Theorem states that if \(a < b\) and \(f\colon [a, b] \to \R\) is continuous on \([a, b]\) and differentiable on \((a, b)\), then there exists \(a < b < c\) such that \[ f(a) - f(b) = f'(c)(a - b). \]
The image of a Cauchy sequence under a uniformly continuous function is a Cauchy sequence.
Continuous function on compact metric spaces are uniformly continuous.
Let \(f\) be a continuous function from a bounded subset \(S\) of a Euclidean space to a Euclidean space. Then \(f\) is uniformly continuous iff for every point \(p \in \partial S\), the limit \(\lim_{x \to p} f(x)\) exists.
A function \(f\colon \R \to \R\) is called periodic if there exists \(P > 0\) such that \(f(x + P) = f(x)\) for all \(x \in \R\). The number \(P\) is called the period of the function \(f\).
Every periodic continuous function \(f\colon \R \to \R\) is uniformly continuous.