convergence in R

**wopashui** · December 5th 2011, 11:58 AM

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.

**girdav** · December 5th 2011, 12:34 PM

$|x_k^{(n)}-x_k|\leq |x^{(n)}-x|$ 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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