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Math Help - limit proof.

  1. #1
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    limit proof.

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

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

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

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

    |(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| \longrightarrow |(x_n-x) +(y_n-y)|

    |(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| \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 (x_n  + y_n ) \to A\;\& \;(x_n  - y_n ) \to B
    Let z_n  = \frac{{(x_n  + y_n ) + (x_n  - y_n )}}{2} then
    \begin{gathered}<br />
  \left| {z_n  - \frac{{\left( {A + B} \right)}}<br />
{2}} \right| \hfill \\<br />
   = \left| {\frac{{(x_n  + y_n ) + (x_n  - y_n )}}<br />
{2} - \frac{{\left( {A + B} \right)}}<br />
{2}} \right| \hfill \\<br />
   \leqslant \left| {\frac{{(x_n  + y_n )}}<br />
{2} - \frac{A}<br />
{2}} \right| + \left| {\frac{{(x_n  - y_n )}}<br />
{2} - \frac{B}<br />
{2}} \right| \hfill \\ <br />
\end{gathered}
    As noted \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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