How do I prove if $\lim a_{2n}=L$ and $\lim a_{2n+1}=L$ then $a_n$ converges and $\lim a_n =L$?
$(z_{2n})_{n\in \mathbb{N}}$ and $(z_{2n+1})_{n\in \mathbb{N}}$ converges to same limit. Show $(z_n)_{n\in \mathbb{N}}$ converges to the same limit.