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Thread: limit proof.

  1. #1
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    limit proof.

    Let $\displaystyle (x_n)_{n \in \mathbb{N}}$ and $\displaystyle (y_n)_{n \in \mathbb{N}}$ be two sequences in $\displaystyle \mathbb{R}$ such that $\displaystyle (x_n+y_n)_{n \in \mathbb{N}}$ and $\displaystyle (x_n-y_n)_{n \in \mathbb{N}}$ both of which converge. Show that $\displaystyle (x_n)_{n \in \mathbb{N}}$ and $\displaystyle (y_n)_{n \in \mathbb{N}}$ converge.

    Hint: let $\displaystyle x_n = \frac{(x_n+y_n)+(x_n-y_n)}{2}$

    I'm able to show that if $\displaystyle (x_n)_{n \in \mathbb{N}}$ and $\displaystyle (y_n)_{n \in \mathbb{N}}$ converge then $\displaystyle (x_n+y_n)_{n \in \mathbb{N}}$ and $\displaystyle (x_n-y_n)_{n \in \mathbb{N}}$ both of which converge.

    for this I'm not sure if it's:

    $\displaystyle |(x_n+y_n)-(x+y)|< \epsilon \longrightarrow \left|\left(\frac{(x_n+y_n)+(x_n-y_n)}{2} +y_n\right) -(x+y)\right|$ $\displaystyle \longrightarrow |(x_n-x) +(y_n-y)|$

    $\displaystyle |(x_n-y_n)-(x-y)|<\epsilon \longrightarrow \left|\left(\frac{(x_n+y_n)+(x_n-y_n)}{2} -y_n\right) -(x-y)\right| $ $\displaystyle \longrightarrow |(x_n-x) -(y_n+y)|$

    if I use the hint I'm back to were I started, which is basically at the beginning, and have no clue how to precede.
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  2. #2
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    Say that $\displaystyle (x_n + y_n ) \to A\;\& \;(x_n - y_n ) \to B$
    Let $\displaystyle z_n = \frac{{(x_n + y_n ) + (x_n - y_n )}}{2}$ then
    $\displaystyle \begin{gathered}
    \left| {z_n - \frac{{\left( {A + B} \right)}}
    {2}} \right| \hfill \\
    = \left| {\frac{{(x_n + y_n ) + (x_n - y_n )}}
    {2} - \frac{{\left( {A + B} \right)}}
    {2}} \right| \hfill \\
    \leqslant \left| {\frac{{(x_n + y_n )}}
    {2} - \frac{A}
    {2}} \right| + \left| {\frac{{(x_n - y_n )}}
    {2} - \frac{B}
    {2}} \right| \hfill \\
    \end{gathered} $
    As noted $\displaystyle \left( {x_n } \right) = \left( {z_n } \right) \to \frac{{\left( {A + B} \right)}}{2}$
    You can fill in the details and finish.
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