(d3.24) Let $(a_{n})$ be a sequence of real numbers and let $n_{1} < n_{2} < \dots < n_{k} < \dots$ be a strictly increasing sequence of real numbers. Then the sequence $(a_{n_{k}})$ given by $(a_{n_{1}}, a_{n_{2}}, \dots, a_{n_{k}}, \dots)$ is called subsequence of $(a_{n})$
if $(a_{n_{k}})$ and $(a_{m_{k}})$ are subsequece of $(a_{n})$ , we say that they cover $(a_{n})$ if ${n_{k}} \cup {m_{k}} = N$

Theorems

every subsequence of a bounded sequence is bounded
(3.25) if $(a_{n})$ converges to $x$ (or tends to $\pm \infty$ ) then every subsequence of $(a_{n})$ is also converges to $x$ (or $\pm \infty$ )
(3.32) (BW) Bolzano–Weierstrass theorem - Every bounded sequence has a convergent subsequence
(3.33) Every sequence has a convergent (or tends to infinity) subsequence
(q3.48) Monotone Subsequence Theorem - Every sequence has a monotonic subsequence

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