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Thread: convergence in R

  1. #1
    Senior Member
    Jan 2010

    convergence in R

    Let $\displaystyle (x^{(n)})^\infty_{n=1}= ((x_k^{(n)})_{k=1}^{\infty})_{n=1}^{\infty} $be a sequence of elements of $\displaystyle l_1$. Prove that if $\displaystyle (x^{(n)})_{n=1}^{\infty}$ converges in $\displaystyle l_1$ to $\displaystyle x=(x_k)_{k=1}^{\infty}$, then for every $\displaystyle K \in N$, the sequence $\displaystyle (x_k^{(n)})_{n=1}^{\infty}$ converges in $\displaystyle R$ to $\displaystyle x_k$. 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

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