(1)

(a) Let $\displaystyle h_n(x)=\frac{sin(nx)}{n}.$ Show that $\displaystyle h_n \rightarrow 0$ uniformly on $\displaystyle \mathbb{R}.$ At which points does the sequence of derivatives $\displaystyle h'_n$ converge?

(b) Modify this example to show that it is possible for a sequence $\displaystyle (f_n)$ to converge uniformly but for $\displaystyle (f'_n)$ to be unbounded.

(2) Consider the sequence of functions defined by $\displaystyle g_n(x)= \frac{x^n}{n}$.

(a) Show $\displaystyle (g_n)$ converges uniformly on$\displaystyle [0,1]$ and find $\displaystyle g=lim$ $\displaystyle g_n$. Show that $\displaystyle g$ is differentiable and compute $\displaystyle g'(x)$ for all $\displaystyle x \in [0,1].$

(b) Now show that $\displaystyle (g'_n)$ converges on [0,1]. Is the convergence uniform? Set $\displaystyle h = lim$ $\displaystyle g'_n$ and compare $\displaystyle h$ and $\displaystyle g'$. Are they the same?

Thanks for any and all help!