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.
Best Answer
Every sequence has at least one limit point because the limit superior and limit inferior are examples of limit points of the sequence. The superior and inferior limits may coincide however (if and only if the sequence converges) so hence "at least one" rather than "at least two".