product of continuous functions is a continuous function

["jennifer"]

This is what i have so far:

Let f be continuous at c and let g be continuous at c

Since f is continuous as c then for all e>0 there is a d_1>0 s.t.
[x-c]<d_1 whenever [f(x)-f(c)]< e/2

and similarly for g, for all e>0 there is a d_2 s.t. [x-c]<d_1 whenever
[f(x)-f(c)]< e/2

I then came up with:

f(x) g(x) - f(c) g(c) |
                            = | f(x) g(x) - f(x) g(c) + f(x) g(c) -
f(c) g(c) |
                             = | f(x) | | g(x) - g(c) | + | g(c) | |
f(x) - f(c) | 

I am stuck here.

What can i pick for e ?