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Thread: Subsequences

  1. #1
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    Subsequences

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

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

    I don't know what to do now.
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  2. #2
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    Re: Subsequences

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

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

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

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

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

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

    Quote Originally Posted by dwsmith View Post
    Why is $\displaystyle N=2(N_1+N_2)$
    First of all, it insures absolutely that $\displaystyle N>N_1~\&~N>N_2$.
    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 $\displaystyle n\text{ is odd}$ we have $\displaystyle k = \left\lfloor {\frac{n}{2}} \right\rfloor > N_2 $ and $\displaystyle z_n=y_k$.
    Can you also explain this?
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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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