# Math Help - Help proving convergence

1. ## 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

2. 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

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