SAMPLE EXAM IN REAL ANALYSIS

Do as many as you can. Write down all details for partial credits.

1. Give the complete definitions of "metric" and "metric space".

2. What does it mean to say that a subset of a metric space is "open"?

3. Show that the intersection of two open subsets of a metric space is open.

4. What is a "limit point" of a subset of a metric space?

5. If $A$ and $B$ are subsets of a metric space, is it true that $(A\cap B)'=A'\cap B'$? Here we denote by $E'$ the set of limit points of $E$ where $E$ is any subset of the metric space.

6. State each of these definitions:

(a) Convergent sequence in a metric space.

(b) Cauchy sequence in a metric space.

(c) Complete metric space.

7. Let $\{a_n\}$ and $\{b_n\}$ be a sequences of real numbers, with $a_n\le b_n$ for all $n$. Prove that $\lim_{n\rightarrow \infty} sup\ a_n\le \lim_{n\rightarrow \infty} sup\ b_n.$

8. Let $f:(0,1)\rightarrow {\bf R}, \ f(x)=\frac{1}{x}$. Show that $f$ is not uniformly continuous.

9. Is the interval $[0,1)$ a complete metric space using the standard metric on the real line? Explain your answer.

10. Is the interval $[0,1]$ a complete metric space using the standard metric on the real line? Explain your answer.

11. Give an example of a complete metric space that is not compact. Explain your answer.

12. Show that a compact subset of a metric space is closed.

13. State the Heine-Borel Theorem.

14. Let a sequence be defined by $x_1=0$ and $x_{n+1}=\sqrt{2+x_n}$. Prove $\{x_n\}$ is bounded above, increasing, and converges to $2$.

15. Give an example of a subset of a metric space that is closed and bounded but is not compact.

16. State the Blozano-Weierstrass Theorem.

17. Show that if $C$ is a connected subset of a metric space and $C\subset D \subset \overline C$, then $D$ is also connected.

18. Let $X$ and $Y$ be metric spaces, $p\in X$ and $f:X\rightarrow Y$. What does it mean to say that "$f$ is continuous at $p$"?

19. Let $X$ and $Y$ be metric spaces, $f:X\rightarrow Y$ continuous, and $C\subset X$. Show that if $C$ is connected, then $f(C)$ is also connected.

20. Let $f: {\bf R}^3\rightarrow {\bf R}^2$ be defined by

$$f_z(x,y,z) =x^2 + y^2 - 5z,\ \ f_2(x, y, z)=2x+y-2z.$$

Notice that $f(3.4.5)=(0,0)$. Use the implicit function theorem to prove that the equation $f(x, y, z)=(0,0)$ can be solved for $x, y$ as functions of $z$ in a neighborhood of $(3,4,5)$. Compute $\frac{dx}{dz}$ at $(3,4,5)$.

21. If $\sum^\infty_{j=1} a_j$ converges where $a_j>0$, prove that $\sum^\infty_{j=1} a^2_j$ convergers.

22. If $f$ is a bounded real-valued function on $[a, b]$ and $\alpha$ a monotone increasing function on $[a, b]$, define the upper and lower Riemann-Stieltjes integrals of $f$ on $[a, b]$ with respect to $\alpha$. What does it mean to say that $f\in R(\alpha)$ on $[a, b]$?

23. Let $\alpha$ be monotone increasing on $[a, b]$ and $f$ and $g$ two bounded real-valued functions on $[a, b]$. Show that if $f\in R(\alpha)$ and $g\in R(\alpha)$ on $[a, b]$, then $f+g\in R(\alpha)$ and $\int^b_a (f+g)d\alpha=\int^b_a f d \alpha + \int^b_a d \alpha$.

24. Let

$$\alpha(x)= \left\{ \begin{array}{ll} & 1, \ 0\le x\le 2,\\ & 2, \... ... & x+1, \ 1<x\le 3, \\ & x+2,\ 3<x\le 5.\\ & x+3, \ 5<x\le 6 \end{array}\right.$$

is $f\in R(\alpha)$ on $[0, 6]$? Give a complete explanation of your answer.