## Real Analysis Exam

Reading time: about 4 minutes (937 words).

## 1994 Nov 16

Instructions. Masters students: Do any 5 problems. Ph.D. students: Do any 6 problems.

1. Let $E$ be a normed linear space. Show that $E$ is complete if and only if, whenever $\sum_1^\infty |x_n | < \infty$, then $\sum_1^\infty x_n$ converges to an $s\in E$.

2. Let $f_n$ be a sequence of real continuous functions on a compact Hausdorff space $X$. Show that if $f_1 \geq f_2 \geq f_3 \geq \cdots$, and $f_n(x) \rightarrow 0$ for all $x\in X$, then $f_n \rightarrow 0$ uniformly.

3. Let $f$ be integrable on the real line with respect to Lebesgue measure. Evaluate $$\lim_{n\rightarrow \infty} \int_{-\infty}^\infty f(x-n) \left(\frac{x}{1+|x|}\right)\ dx.$$ Justify all steps.