\( \newcommand{\cat}[1] {\mathrm{#1}} \newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})} \newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}} \newcommand{\multiBetaReduction}[0] {\twoheadrightarrow_{\beta}} \newcommand{\betaReduction}[0] {\rightarrow_{\beta}} \newcommand{\betaEq}[0] {=_{\beta}} \newcommand{\string}[1] {\texttt{"}\mathtt{#1}\texttt{"}} \newcommand{\symbolq}[1] {\texttt{`}\mathtt{#1}\texttt{'}} \newcommand{\groupMul}[1] { \cdot_{\small{#1}}} \newcommand{\inv}[1] {#1^{-1} } \newcommand{\bm}[1] { \boldsymbol{#1} } \require{physics} \require{ams} \)

Subsequences and limits, proposition

Subsequences and limits

Let \( (a_n)_{n=0}^{\infty} \) be a sequence of real numbers, and let \( L \) be a real number. Then the the following two statements are logically equivalent:

• The sequence \( (a_n)_{n=0}^{\infty} \) converges to \( L \).
• [...]

Subsequences related to limit points

Let \( (a_n)_{n=0}^{\infty} \) be a sequence of real numbers, and let \( L \) be a real number. Then the following two statements are logically equivalent:

• [...]
• There exists a subsequence of \( (a_n)_{n=0}^{\infty} \) which converges to \( L \).