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Math Help - convergence of sequence and subsequence

  1. #1
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    convergence of sequence and subsequence

    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'.
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  2. #2
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    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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