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Math Help - Subsequences

  1. #1
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    Subsequences

    Suppose \{x_n\}\to x_0 and \{y_n\}\to x_0. Define a sequence \{z_n\} as follows: z_{2n}=x_n and z_{2n-1}=y_n. Prove that \{z_n\} converges to x_0.

    Let \epsilon >0. Then \exists N_1, \ N_2\in\mathbb{N} such that for n\geq N_1, \ N_2 we have |x_n-x_0|<\epsilon and |y_n-x_0|<\epsilon.

    I don't know what to do now.
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  2. #2
    Super Member girdav's Avatar
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    Re: Subsequences

    Let N:=\max(N_1,N_2). For n\geq N we have |z_{2n}-x_0|\leq \varepsilon and |z_{2n-1}-x_0|\leq \varepsilon hence if k\geq 2N-1 we have |z_k-x_0|\leq \varepsilon.
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  3. #3
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    Re: Subsequences

    Quote Originally Posted by dwsmith View Post
    Suppose \{x_n\}\to x_0 and \{y_n\}\to x_0. Define a sequence \{z_n\} as follows: z_{2n}=x_n and z_{2n-1}=y_n. Prove that \{z_n\} converges to x_0.

    Let \epsilon >0. Then \exists N_1, \ N_2\in\mathbb{N} such that for n\geq N_1, \ N_2 we have |x_n-x_0|<\epsilon and |y_n-x_0|<\epsilon.
    Let N=2(N_1+N_2). If n\ge N then if n\text{ is odd} we have k = \left\lfloor {\frac{n}{2}} \right\rfloor  > N_2 and z_n=y_k.

    Use a similar idea if n\text{ is even}.
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    Re: Subsequences

    Quote Originally Posted by Plato View Post
    Let N=2(N_1+N_2). If n\ge N then if n\text{ is odd} we have k = \left\lfloor {\frac{n}{2}} \right\rfloor  > N_2 and z_n=y_k.

    Use a similar idea if n\text{ is even}.
    Why is N=2(N_1+N_2)
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    Re: Subsequences

    Quote Originally Posted by dwsmith View Post
    Why is N=2(N_1+N_2)
    First of all, it insures absolutely that N>N_1~\&~N>N_2.
    Therefore, we can use anyone of the statements already restricted.
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    Re: Subsequences

    Quote Originally Posted by Plato View Post
    then if n\text{ is odd} we have k = \left\lfloor {\frac{n}{2}} \right\rfloor  > N_2 and z_n=y_k.
    Can you also explain this?
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  7. #7
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    Re: Subsequences

    Quote Originally Posted by dwsmith View Post
    Can you also explain this?
    You do the mathematics.
    Just take many cases.
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