Problem 4.2.1 concerns the uniform convergence of the sequence of functions given by fn(x)=(sin nx)/n. As you can see, the whole function shrinks in a uniform manner towards the x axis.
Problem 4.2.2 concerns the lack of uniform convergence of the sequence of functions given by fn(x)=(sin nx)/nx. As you can probably see, the function value tends to zero everywhere, but more slowly near x=0. Since the values always approach 1 as x approaches zero, the convergence is not uniform.
Harald Hanche-Olsen Updated: 2006–10–23 17:37