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Thread: convergence in R

  1. #1
    Senior Member
    Jan 2010

    convergence in R

    Let  (x^{(n)})^\infty_{n=1}= ((x_k^{(n)})_{k=1}^{\infty})_{n=1}^{\infty} be a sequence of elements of l_1. Prove that if (x^{(n)})_{n=1}^{\infty} converges in l_1 to x=(x_k)_{k=1}^{\infty}, then for every K \in N, the sequence (x_k^{(n)})_{n=1}^{\infty} converges in R to x_k. Show by example that the converse is not true.
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  2. #2
    Super Member girdav's Avatar
    Jul 2009
    Rouen, France

    Re: convergence in R

    |x_k^{(n)}-x_k|\leq |x^{(n)}-x| and taking x^{(k)}=e^{(k)} where e_n^{(k)}=\begin{cases}1&\mbox{ if }n=k\\0&\mbox{ otherwise}\end{cases} we get a counter-example for the converse.
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