First, is d well defined (in other words, is that sup finite)? —Yes, because the negative exponentials never take a value greater than 1.

Next, are the axioms satisfied?

(1) Is d(f,g)≥0? —Yes, obviously.

(2) If d(f,g) = 0, does that imply that f = g? (I'll leave that one to you.)

(3) Is d(g,f) = d(f,g)? —Yes, obviously.

(4) Is d(f,h) ≤ d(f,g) + d(g,h)? (Use the ordinary triangle inequality for real numbers here.)

To see that neither of these series of functions converge, notice that in both cases the pointwise limit of the series is the function f(x)=x^2. Then ask yourself whether the difference between the n'th function in the series and the limit function can be made smallfor all x in Rby taking n large enough.

On the other hand, and both tend to 0 as n→∞.