Pointwise Convergence Vs Uniform Convergence?

Suppose {f_n} is a sequence of functions defined on a set X.

f_n converges to a function f "pointwise" if for each x in X, (f_n)(x) converges to f(x). That is, if we just look at the sequence of functions at a single point x in X, the values of the f_n at that point go to the value of f at that point. Simple. It's exactly what you'd think "point-wise" means: convergence happens at each individual point.

f_n converges to f "uniformly" if for each e>0, there is a natural number N such that for all n>=N, |(f_n)(x) - f(x)| < e for ALL points points x in X. In other words, for uniform convergence, if we choose any level of closeness e>0, we can make the sequence of functions f_n that close to the limit function everywhere on X at once. For point-wise convergence, we only had to make the functions close to the limit at each individual point x by itself. For uniform convergence, we have to make the functions close to the limit function for all the points at once.