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Math Help - Help proving convergence

  1. #1
    Junior Member
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    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
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by roporte View Post
    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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