Another one that I'm kind of uncomfortable with my solution:

B(X) vector space of all bounded functions in X. Show that it's going to be defined under the sup norm:

d(f,g) = sup (mod(f(x)-g(x))

What I did was to show that every Cauchy sequence in B(X) converges to an f, (so mod(f-fn) gets arbitrarily small with n --> infinity);

and the terms fn and fm get arbitrarily close to each other with a high n. So for a high enough n, we have

mod(f) <= mod(fn) + mod(f - fn) < E, for an arbitrary E, so f , the limit of the cauchy sequence, is a bounded function, belonging to B(X). Hence, the space is complete.

I feel that I'm missing something here, since i didn't use the sup norm directly.