Thread: convergence pointwise

Give an example of a sequence of bounded functions f_n converging pointwise to an unbounded function f.
Prove that if all functions f_n are bounded and f_n converge to f uniformly on a set I then f is bounded.

Can anybody give me some hints how to prove this question please?

Thanks for your time indeed

Originally Posted by knguyen2005

Give an example of a sequence of bounded functions f_n converging pointwise to an unbounded function f.

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

Originally Posted by knguyen2005

Prove that if all functions f_n are bounded and f_n converge to f uniformly on a set I then f is bounded.

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.