Sequential language is the working grammar of real analysis. Limits, subsequences, limsup, and asymptotic notation are the tools used to control approximation and extract structure from infinite objects.
Epsilon definition of convergence
Convergence means all sufficiently late terms lie in every prescribed neighborhood of the limit.
Definition. A sequence \(x_n\) converges to \(x\), written \(x_n\to x\), if for every \(\varepsilon>0\) there is \(N\) such that \(n\ge N\Rightarrow |x_n-x|\lt \varepsilon\).
Proposition. Limits are unique and every convergent sequence is bounded.
Proof. If \(x_n\to x\) and \(x_n\to y\), then \(|x-y|\le |x-x_n|+|x_n-y|\), and the right side can be made arbitrarily small, so \(x=y\). Boundedness follows by bounding the tail in a unit neighborhood of the limit and taking the maximum of finitely many initial absolute values.
Proposition (algebra and order of limits). If \(x_n\to x\), \(y_n\to y\), and \(c\in\mathbb R\), then \[ x_n+y_n\to x+y,\qquad cx_n\to cx,\qquad x_ny_n\to xy. \] If \(y\ne0\), then \(x_n/y_n\to x/y\). Moreover, if \(x_n\le y_n\) eventually, then \(x\le y\).
Proof. The first two assertions follow directly from the triangle inequality. Since a convergent sequence is bounded, say \(|x_n|\le M\), \[ |x_ny_n-xy|\le |x_n|\,|y_n-y|+|y|\,|x_n-x| \le M|y_n-y|+|y|\,|x_n-x|\longrightarrow0. \] When \(y\ne0\), eventually \(|y_n|\ge |y|/2\), and hence \[ \left|\frac1{y_n}-\frac1y\right| =\frac{|y_n-y|}{|y_ny|} \le \frac{2}{|y|^2}|y_n-y|\longrightarrow0. \] The quotient rule now follows from the product rule. Finally, if \(x>y\), choose \(\varepsilon=(x-y)/3\). Eventually \(x_n>x-\varepsilon>y+\varepsilon>y_n\), contradicting the eventual inequality.
Theorem (monotone convergence). An increasing real sequence converges if and only if it is bounded above; in that case \[ \lim_{n\to\infty}x_n=\sup_{n\ge1}x_n. \] The analogous statement holds for decreasing sequences and infima.
Proof. A convergent sequence is bounded. Conversely, let \(x_n\) be increasing and bounded above, and put \(s=\sup_nx_n\). For every \(\varepsilon>0\), \(s-\varepsilon\) is not an upper bound, so \(x_N>s-\varepsilon\) for some \(N\). Monotonicity gives \(s-\varepsilon<x_N\le x_n\le s\) whenever \(n\ge N\), proving \(x_n\to s\).
Subsequences and cluster points
Subsequences reveal the possible limiting behaviors hidden inside a sequence.
Definition. A subsequence has the form \(x_{n_k}\) with \(n_1\lt n_2\lt \cdots\). A point \(x\) is a cluster point if some subsequence converges to \(x\).
Theorem. A sequence \(x_n\) converges to \(x\) if and only if every subsequence has a further subsequence converging to \(x\).
Proof. If \(x_n\to x\), every subsequence also converges to \(x\). Conversely, if \(x_n\) does not converge to \(x\), then for some \(\varepsilon>0\) there are infinitely many \(n\) with \(|x_n-x|\ge\varepsilon\). Those terms form a subsequence with no further subsequence converging to \(x\).
Limsup and liminf
Limsup and liminf record the eventual upper and lower envelopes.
Definition. For a real sequence, \(\limsup x_n=\inf_N\sup_{n\ge N}x_n\) and \(\liminf x_n=\sup_N\inf_{n\ge N}x_n\), with extended real values allowed.
Writing \(s_N=\sup_{n\ge N}x_n\) and \(i_N=\inf_{n\ge N}x_n\) makes the construction transparent: \(s_N\) decreases and \(i_N\) increases. Thus their limits exist in the extended line and equal the displayed infimum and supremum.
Theorem (subsequential characterization). If \((x_n)\) is bounded, then \(\limsup x_n\) is its largest subsequential limit and \(\liminf x_n\) is its smallest subsequential limit. In particular, both are attained as limits of subsequences.
Proof. Let \(s=\lim s_N\), where \(s_N=\sup_{n\ge N}x_n\). For each \(k\), choose \(N_k\) so large that \(s_{N_k}<s+1/k\), and then choose \(n_k\ge\max\{N_k,n_{k-1}+1\}\) such that \(x_{n_k}>s_{N_k}-1/k\). Because \(s\le s_{N_k}\), \[ s-\frac1k<x_{n_k}<s+\frac1k, \] so \(x_{n_k}\to s\). If \(x_{m_k}\to\ell\), then for every fixed \(N\), eventually \(m_k\ge N\), hence \(x_{m_k}\le s_N\). Passing to the limit gives \(\ell\le s_N\), and then \(\ell\le s\). The liminf statement follows by applying this argument to \(-x_n\).
Proposition. A bounded sequence converges if and only if \(\limsup x_n=\liminf x_n\), and the common value is the limit.
Proof. If \(x_n\to x\), then every tail eventually lies in \((x-\varepsilon,x+\varepsilon)\), so both envelopes lie within \(\varepsilon\) of \(x\). Conversely, if the two envelopes equal \(x\), then for every \(\varepsilon>0\) the tail supremum is \(\lt x+\varepsilon\) and the tail infimum is \(>x-\varepsilon\), forcing all sufficiently late terms into that interval.
Cauchy criterion
Completeness allows convergence to be checked internally.
Theorem. A real sequence converges if and only if for every \(\varepsilon>0\) there is \(N\) such that \(m,n\ge N\Rightarrow |x_m-x_n|\lt \varepsilon\).
Proof. The forward implication follows from the triangle inequality. For the reverse implication, Cauchy sequences are bounded and therefore have convergent subsequences. If \(x_{n_k}\to x\), the Cauchy property gives \(|x_n-x|\le |x_n-x_{n_k}|+|x_{n_k}-x|\) for large \(n,k\).
Asymptotic notation
Asymptotic notation packages comparison estimates without naming constants at every line.
Definition. For an eventually nonzero comparison sequence \(b_n\), write \[ \begin{aligned} a_n&=O(b_n) &&\Longleftrightarrow && |a_n|\le C|b_n|\ \text{eventually for some }C<\infty,\\ a_n&=o(b_n) &&\Longleftrightarrow && a_n/b_n\to0,\\ a_n&\asymp b_n &&\Longleftrightarrow && 0<c\le |a_n/b_n|\le C<\infty\ \text{eventually},\\ a_n&\sim b_n &&\Longleftrightarrow && a_n/b_n\to1. \end{aligned} \]
Proposition (calculus of orders). If \(a_n=O(b_n)\) and \(c_n=O(d_n)\), then \(a_nc_n=O(b_nd_n)\) and \(a_n+c_n=O(|b_n|+|d_n|)\). If \(a_n=o(b_n)\) and \(c_n=O(d_n)\), then \(a_nc_n=o(b_nd_n)\). Also, \(a_n\sim b_n\) implies \(a_n=b_n+o(b_n)\).
Proof. Each assertion follows after dividing by the proposed comparison scale. For example, \[ \frac{|a_nc_n|}{|b_nd_n|} =\frac{|a_n|}{|b_n|}\frac{|c_n|}{|d_n|}, \] which is bounded in the \(O\)-\(O\) case and converges to zero in the \(o\)-\(O\) case.
Remark (uniformity in a parameter). A statement such as \(r_n(\theta)=O(a_n)\) is ambiguous until the dependence of the constant is specified. Uniformly for \(\theta\in\Theta\) means \[ \sup_{\theta\in\Theta}\frac{|r_n(\theta)|}{|a_n|}\le C \] eventually with one constant \(C\). This distinction is essential when an asymptotic expansion is optimized over, integrated over, or evaluated at a random parameter.