Sum of two function sequences [closed]

Let (fn) and (gn) be two function sequences that converge uniformly to the functions f and g respectively. Let a and b two real numbers.

Then the function sequence (a.fn + b.gn) converges uniformly to the function a.f+b.g

This is a theorem in Analyse courses. I wonder why we need uniform convergence here. Can someone give me an example where f and g are respectively limites (simple limites, not uniform limites) of the sequences (fn) et (gn) but the function sequence (a.fn + b.gn) don't converge uniformly to the function a.f+b.g