Is Uniform Convergence Implying Boundedness of the Limit Function?

[dracond]

Homework Statement

Just trying to get feel for uniform convergence and it's relationship to boundedness. If a sequence of functions (fn) converges uniformly to f and (fn) is a sequence of bounded functions, is f also bounded?

Homework Equations

The Attempt at a Solution

 

[Dick]

If the convergence is uniform then for all e there is an N such that |fn(x)-f(x)|<e for all n>=N. If fN is bounded and f unbounded, how can this be?