1. ## Help proving convergence

Please, could you help me with this demonstration?

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

A) If $\displaystyle \lambda_n$ is a scalar sequence that converges to $\displaystyle \lambda$ prove that $\displaystyle \lambda_n$$\displaystyle x_n->\displaystyle \lambda x b) Let \displaystyle z_n=(x_1+..+x_n)/n. Prove that \displaystyle z_n->x Thanks a lot 2. Originally Posted by roporte A) If \displaystyle \lambda_n is a scalar sequence that converges to \displaystyle \lambda prove that \displaystyle \lambda_n$$\displaystyle x_n$->$\displaystyle \lambda x$

Express

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

and take norms

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