Suppose $(x_n)_{n=1}^\infty$ is a sequence in $\mathbb R$, and that $L_k$ are real numbers with $\lim_{k\to\infty}L_k=L$. If for each $k\geq 1$, there is a subsequence of $(x_n)_{n=1}^\infty$ converging to $L_k$, show that some subsequence converges to $L$. HINT: Find an increasing sequences $n_k$ such that $|x_{n_k}-L|<1/k$.
Can someone tell me what $L_k$ actually are? Is that a sequence or is it something else? I thought it was a sequence first, but the following sentence suggests it isn't (you can't converges to a sequence)