# Every sequence has at least one limit point

This is, of course, easily proven with the help of the Bolzano-Weierstrass theorem. However, going through the lecture notes in my class, the B-W theorem is only introduced after the claim

Every sequence has at least one limit point

is introduced, which itself is introduced as a corollary of:

Theorem:

1) The element $a\in\mathbb R\cup\{{-\infty,\infty}\}$ is a limit point of a sequence $\{X_n\}$ iff exists its subsequence $\{X_{k_n}\}$ converging to $a$.

2) Limit inferior of $\{X_n\}$ is the lowermost limit point of the sequence.

3) Limit superior of $\{X_n\}$ is the uppermost limit point of the sequence.

(Hopefully the idea is clear, this is only my own translation as the text is not english)

I don't quite see how does that imply that "every sequence has at least one limit point" and therefore would be grateful for any help, thanks.

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@DahnJahn Yes, the limit superior and limit inferior of a sequence always exist, because they are defined to be the limits of $a_n=\sup_{k\geq n} \{ x_k \}$ and $b_n=\inf_{k\geq n} \{ x_k \}$ respectively, and both $a_n$ and $b_n$ are monotone (make sure you think about why) thus their limits exist. – Ragib Zaman Dec 10 '12 at 13:32