I need some help with this proof. A subsequential limit is a limit of a subsequence.
Suppose $a_n$ is a sequence with $\{ L_1, L_2, ...\}$ subsequential limits. Suppose $L_n \to L$. Prove that $L$ is a subsequential limit of $a_n$.
Proof:
We know (1): $\forall_{\epsilon > 0} \exists_{N_0} s.t \forall_{n>N_0} \implies |L_n - L| < \epsilon$
(2) For i = 1 to ... $\forall_{\epsilon_i > 0} \exists_{N_i} s.t \forall_{n_i > N_i} \implies |a_{n_i} - L_i| < \epsilon$
- Can I just take my subsequence terms from the interval $|L_n - L| < \epsilon$ and then conclude by the definition of the limit that $L$ is subsequential limit?
- Other than that I am unsure of a general strategy of how to construct proofs with subsequences