Show that $R_n(x)\to 0$ as $n\to \infty$.

Question

Consider the sequence space $\ell^2$ with its usual norm and the sequence of operators $R_n:\ell^2\to\ell^2$, such that for $x\in\ell^2$

$$R_n(x)=(0,0,\dots,0,x_{n+1},x_{n+2},\dots)$$

I am trying to show that $R_n(x)\to 0$ in $\ell^2$ as $n\to \infty$.

In the solution to the problem it says that since $\|x\|_{\ell^2}^2=\sum_{i=1}^\infty|x_i^2|\lt\infty$ then $\sum_{i=n+1}^\infty|x_i|^2\to0$ as $n\to\infty$. How exactly is it that the second sum tends to zero?

Answer 1

This holds since the sum converges:

Set $s_n:=\sum_{i=1}^n|x_i^2|$ then obviously $s_n$ converges to $s:=\sum_{i=1}^\infty|x_i^2|$ this means that $s-s_n \to 0$ but $s-s_n=\sum_{i=n+1}^\infty|x_i^2|$ holds.

But behold, $R_n$ does not converge to $0$ in operator-norm, since for every $n$ you find a sequence $x$ with $\|R_n(x)\|_{\ell^2} \geq 1$ so $R_n$ only converges pointswise towards $0$.