Convergence of a sequence whose even and odd subsequences converge [duplicate]
Suppose $\{a_n\}$ is a sequence such that the subsequences $\{a_{2n−1}\}$ and $\{a_{2n}\}$ converge to the same limit, say $a$. Show that $\{a_n\}$ also converges to $a$.
Hint: Take any $\epsilon>0.$ Since $a_{2n-1}\to a,$ then there is an $N_1$ such that $|a_{2n-1}-a|<\epsilon$ whenever $n>N_1.$ Likewise, there is an $N_2$ such that $|a_{2n}-a|<\epsilon$ whenever $n>N_2.$ How can we use $N_1$ and $N_2$ to come up with an $N$ such that $|a_n-a|<\epsilon$ whenever $n>N$?