Prove that L is partial limit of $a_n$

Question

Let $a_n$ bounded sequence and $b_n$ sequence of all partial limits of $a_n$

$\lim _{n\to \infty }\left(b_n\right)= L$

Prove that L is partial limit of $a_n$ ?

By Bolzano–Weierstrass theorem there is a subsequence $b_{n_k} \to L$

I thinking to prove that $b_{n_k}$ is also a subsequence of $a_n$ ..

but I stuck here ! any help how to prove it (if this the correct way to prove )/ if not how to prove it ?

thx

Answer 1

Hint: For each $k$ there exist $n$ such that $|L-b_n| <\frac 1 k$ and there exists $m$ such that $|b_n-a_m| <\frac 1 k$. Write $m$ as $n_k$. We may assume that $n_1 <n_2,...$ (Why?). It follows now that $(a_{n_k})$ converges to $L$.