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.