# convergence of sequence and subsequence

If $\varepsilon > 0$ then from the given $\left[ {\exists N_1 }{n \geqslant N_1 \Rightarrow \quad \left| {a_{2n} - a} \right| < \varepsilon } \right]\,\& \,\left( {\exists N_2 } \right)\left[ {n \geqslant N_2 \Rightarrow \quad \left| {a_{2n + 1} - a} \right| < \varepsilon } \right]$.
So if $k \geqslant N_1 + N_2$ then $k$ is either even or odd so in eith case ${\left| {a_k - a} \right| < \varepsilon }$