Let A,B,C be nonempty sets of real numbers, and let f:A->B and g:B->C be uniformly continuous functions. Prove that the composition gof:A->C is also unfiormly continuous,

Here's what I have. Is it correct?

Let then there is s.t for all x1,y1 if then | so f(x1),f(y1) B. Now let >0 then there exists >0 s.t for all x2,y2 B if |x2-y2|< then |g(x2)-g(y2)|< . So |(gof)(x1)-(gof)(y1)|=|g(f(x1))-g(f(y1))|< .