I have the following problem involving the set $Y$ of infinite sequences that absolutely converge such that,
$$\sum_{i=0}^\infty x_i^2 \lt\infty$$
where $x_i$ is the $i$-th term in the infinite sequence $x$, together with the metric, $d_2$, defined on $Y$ as
$$d_2(x,y)=\left(\sum_{i=0}^\infty |x_i-y_i|^2 \right)^{1/2} $$
I am now to show that $d_2(x,y)$ is always finite for sequences in $Y$. Here are the facts that I need to make use of:
- $\sum_{i=0}^\infty |x_i|\lt\infty$
- For $a,b\in \mathbb R, (a-b)^2\le2(a^2+b^2)$
I'm not sure what I am allowed to do involving the absolute values; am I able to keep the absolute value as it is in the above and use the fact from point 2. directly or should I use the fact that,
$$|x|=\sqrt{x^{2}}\implies |x|^2=x^2$$
to get rid of the absolute value at the start? On that point, I was unsure about "putting it back in" at the end, so to speak. Here is what I have done so far:
$$\left(\sum_{i=0}^\infty |x_i-y_i|^2 \right)^{1/2} \le\left(2\sum_{i=0}^\infty |x_i^2-y_i^2| \right)^{1/2} $$
$$\implies\left(2\sum_{i=0}^\infty |x_i^2-y_i^2| \right)^{1/2}-\left(\sum_{i=0}^\infty |x_i-y_i|^2 \right)^{1/2}\ge 0$$
$$\implies\left(\sum_{i=0}^\infty |x_i+y_i|^2 \right)^{1/2}\ge0$$
$$\implies\left(\sum_{i=0}^\infty |x_i+y_i|^2 \right)^{1/2}=\left(\sum_{i=0}^\infty |x_i|^2 \right)^{1/2}+\left(\sum_{i=0}^\infty |y_i|^2 \right)^{1/2}+\left(\sum_{i=0}^\infty 2|x_i||y_i| \right)^{1/2}$$
Which is true because the $\sqrt{}$ function is monotone increasing. Using the fact of 1. it would follow that each of the terms on the right hand side of the last statement would be $\lt\infty$ and so $d_2(x,y)\lt\infty$ as required.
I know what I have to obtain, and think I got it in the end, although I am not sure about the means to get there.