Infinite sum of squares converges

[rjw5002]

Homework Statement

So, L2 is defined to be the set of all infinite sequences of real numbers, s.t. the sum of their squares converges:
L2 = {x=(x1,...,xn,...) | $$\Sigma$$xi < $$\infty$$}
we have d(x,y) = $$\sqrt{\Sigma (xi-yi)^2}$$

I need to show that this is a metric, starting by showing that if $$xi,yi\in$$ then $$xi-yi\in$$

Homework Equations

definition of a metric

The Attempt at a Solution

so my initial problem was with the first step:
x = (x1,..., xn,...), y = (y1,...,yn,...), and then x - y = (x1 - y1, ...,xn-yn,...)
to show $$xi,yi\in$$, we must show that $$\Sigma (xi -yi) < \infty$$.
First, I broke up the squared quantity to get $$\Sigma (xi^2- 2xi*yi + yi^2)$$.
Carrying the sumation through, $$\Sigma xi^2 < \infty$$ (as for yi^2).
But where I get stuck is how to show or determine that $$\Sigma xi*yi < \infty$$.

So then the first three parts of the definition of a metric ($$d(p,p) = 0, p\rightarrow d(p,q) > 0,$$ and $$d(p,q) = d(q,p)$$) are easy enough to prove assuming $$xi-yi \in L2$$.
I get caught up again in proving that d(p,q) < d(p,r) + d(r,q).
I tried squaring both sides and distributing the summation, but found the resulting right hand side of the equation did not really simplify in a useful way. I'm not really sure where to go from there... any ideas for either of these roadblocks would be greatly appreciated. Thanks.

 

[Dick]

Your missing ingredients are various facets of the 'Cauchy-Schwarz inequality'. Look that up.