TITLE: Bolzano–Weierstrass theorem conclusion
QUESTION [5 upvotes]: Can I conclude from Bolzano–Weierstrass theorem that there is more than one convergent subsequence, or the theorem tells me that there's only one ?
To be more clear, given a bounded sequence $X_n$, not ecessarily converges, can I conclude there are two different subsequences $X_{n_k}$ that converges to $L_1$ and $X_{n_l}$ that converges to $L_2$?

REPLY [3 votes]: An equivalent formulation of the BW theorem is that any bounded sequence has at least one accumulation point.
As noted in the answer of Omnomnomnm we also know that for a convergent sequence, all subsequences converge to the same limit.
So, given two convergent sequences $\{a_n\} \to a$ and $\{b_n\} \to b$ with $a \ne b$ , the sequece $\{c_n\}$ with 
$c_{2k}=a_k$ and $c_{2k+1}=b_k$ is bounded ( because the two starting sequeces are bounded) and has two different accumulation points. 
Genarizing this we can construct a bounded sequence with many accumulation points.