Seems basic but I'm not getting it....

[futurebird]

I'm studying the proof that the l2 metric space is complete and I just don't get something about this proof.

What I know:

1. l2 is the metric space whose elements are sequences that converge when you square each term and sum them. In other words, a sequence, f is in l2 if SUM( |f(k)|^2) < infinity.
2. The norm on this metric space is: (SUM( |f(k)|^2))^0.5 = || f ||2
3. A sequence in the l2 space is a sequence of sequences (confusing isn't it?)
4. To show that the space is complete I need to show that every cauchy sequence in l2 converges to a sequence in l2.

My book says:

Suppose that for e > o there exists n0 such that:

||fn - fm||2 < e when m, n >= n0.

I don't understand how to think about ||fn - fm||2.  How would I find ||fn - fm||2 computationally?

Suppose fn = .1, .01, .001, ...
Suppose fm = .2, .02, .002, ...

Do I do:

||fn - fm||2 = [|(.1^2 + .01^2, .001^2 + ...) - (.2^2 + .02^2 + .002^2 + ...)|]^0.5
||fn - fm||2 = [|(.01 + .001, .0001 + ...) - (.04 + .004 + .0004 + ...)|]^0.5
||fn - fm||2 = [|(.01 + .001, .0001 + ...) - (.04 + .004 + .0004 + ...)|]^0.5
||fn - fm||2 = [ |1/90 - 4/90|]^0.5
||fn - fm||2 = [3/90]^0.5

or

[(.1 - .2)^2 + (.01-.02)^2, + ...]^0.5 ?

I think it's the first way... but for some reason I'm confused. Deeply.