convergence of sequence and subsequence

**sah_mat** · November 8th 2008, 11:58 AM

hi again,can u help me to show
if X(2n) and X(2n+1) convergence to 'a',then show X(n) convergence to 'a'.

**Plato** · November 8th 2008, 12:20 PM

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 ${\left| {a_k - a} \right| < \varepsilon }$

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