Question
If $\left(a_{n_k}\right)$ is a convergent subsequence of $\left(a_n\right)$, show that $\liminf _{n \rightarrow \infty} a_n \leq$ $\lim _{k \rightarrow \infty} a_{n_k} \leq \lim \sup _{n \rightarrow \infty} a_n$.
Step 1
The limit inferior of a sequence \((a_n)\) is defined as: \[ \liminf_{n \to \infty} a_n = \lim_{n \to \infty} \inf_{m \geq n} a_m \] and the limit superior is defined as: \[ \limsup_{n \to \infty} a_n = \lim_{n \to \infty} \sup_{m \geq n} a_m. \] Show more…
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