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Math Help - true or false? similar to Toeplitz theorem

  1. #1
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    true or false? similar to Toeplitz theorem

    y_{n}=a_{1n}x_{1}+a_{2n}x_{2}+\dotsb+a_{nn}x_{n}.

    \lim_{n\to \infty}a_{kn}=0,\ \forall k\in\mathbb N ;

    \forall i, j, i\ne j,\ \lim_{n\to\infty} a_{i}/a_{j}=1 ;

    a_{1k}+a_{2k}+\dotsb+a_{nk}=1 .

    IS \lim_{n\to \infty} x_n=A \Longrightarrow \lim_{n\to \infty}y_n=A ?
    Last edited by CaptainBlack; January 9th 2010 at 12:01 PM. Reason: trying to fix LaTeX
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  2. #2
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    Sorry for typo. The question is:
    IS \lim_{n\to \infty} y_n=A \Longrightarrow \lim_{n\to \infty}x_n=A ?
    Quote Originally Posted by Sbarcoo View Post
    y_{n}=a_{1n}x_{1}+a_{2n}x_{2}+\dotsb+a_{nn}x_{n}.

    \lim_{n\to \infty}a_{kn}=0,\ \forall k\in\mathbb N ;

    \forall i, j, i\ne j,\ \lim_{n\to\infty} a_{i}/a_{j}=1 ;

    a_{1k}+a_{2k}+\dotsb+a_{nk}=1 .

    IS \lim_{n\to \infty} x_n=A \Longrightarrow \lim_{n\to \infty}y_n=A ?
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