true or false? similar to Toeplitz theorem

**Sbarcoo** · January 9th 2010, 06:25 AM

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

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

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

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

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

**Sbarcoo** · January 10th 2010, 02:12 AM

Sorry for typo. The question is:
IS $\lim_{n\to \infty} y_n=A \Longrightarrow \lim_{n\to \infty}x_n=A$ ?

Originally Posted by Sbarcoo

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

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

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

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

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

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