On the open unit interval (0,1), the functions are bounded, but converge pointwise to 1/x, which is unbounded.

In words, if f_n converges uniformly to f then for some n, f_n(x) will be within distance 1 of f(x), for all x. But if |f_n(x)| is bounded by M, then |f(x)| will be bounded by M+1.