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Math Help - Show that the sequence converges to L

  1. #1
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    Show that the sequence converges to L

    Suppose that a_n --> L and b_n --> L. Show that the sequence

    a_1,b_1,a_2,b_2,a_3,b_3,...

    converges to L.
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  2. #2
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    Quote Originally Posted by wopashui View Post
    Suppose that a_n --> L and b_n --> L. Show that the sequence

    a_1,b_1,a_2,b_2,a_3,b_3,...

    converges to L.

    Let \epsilon > 0\Longrightarrow \exists\,N_1,\,N_2\in\mathbb{N}\,\,\,s.t.\,\,\,|a_  n-L|<\epsilon\,\,\forall\,n>N_1\,\,\,and\,\,\,|b_n-L|<\epsilon\,\,\forall\,n>N_2 .

    Let M:=\max(N_1,N_2) , then \forall\,n>M\,,\,\,|x_n-L|<\epsilon , with x_n=a_n\,\,\,or\,\,\,b_n (either one, it doesn't matter), so...

    Tonio
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