# convergence in R

• December 5th 2011, 12:58 PM
wopashui
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.
• December 5th 2011, 01:34 PM
girdav
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.