Help proving convergence

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May 2nd 2011, 09:11 AM
roporte

Help proving convergence

Please, could you help me with this demonstration?

Let X a vectorial space with norm and ${x_n}$ e ${y_n}$ sequences over X where x_n->x and y_n->Y.

A) If $\lambda_n$ is a scalar sequence that converges to $\lambda$ prove that $\lambda_n$ $x_n$ -> $\lambda x$

b) Let $z_n$ =(x_1+..+x_n)/n. Prove that $z_n$ ->x

Thanks a lot
May 2nd 2011, 11:07 AM
FernandoRevilla

Quote:

Originally Posted by roporte

A) If $\lambda_n$ is a scalar sequence that converges to $\lambda$ prove that $\lambda_n$ $x_n$ -> $\lambda x$

Express

$\lambda_nx_n-\lambda x=\lambda (x_n-x)+(\lambda_n-\lambda)x+(\lambda_n-\lambda)(x_n-x)$

and take norms

$\left\|{\lambda_n x_n-\lambda x}\right\|\leq \left |{\lambda}\right | \left\|{x_n-x}\right\|+\ldots$

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