• Work 5 out of 6 problems.
• Each problem is worth 20 points.
• Write on one side of the paper only and hand your work in order.
• Do not interpret a problem in such a way that it becomes trivial.

(1) (a) Let $f _ { n } ( x ) = { \frac { 1 } { 1 + n ^ { 2 } x ^ { 2 } } }$ and $g _ { n } ( x ) = n x ( 1 - x ) ^ { n }$ for $x \in [ 0 , 1 ]$ . Prove that $\left\{ f _ { n } \right\}$ and $\left\{ g _ { n } \right\}$ converge pointwise but not uniformly on $[ 0 , 1 ]$

(b) Are the families $\left\{ f _ { n } \right\}$ , respectively $\left\{ g _ { n } \right\}$ given in part (a) equicontinuous?
Clearly motivate your answer.

(2) Consider the following subset of the metric space $( \mathcal { C } _ { \mathrm { b } } ( [ 0 , 1 ] ) , \rho _ { \infty } )$

$$
A : = \big \{ f \in \mathcal { C } _ { \mathrm { b } } ( [ 0 , 1 ] ) : f ( [ 0 , 1 ] ) \subseteq [ 0 , 1 ] \big \} .
$$

(a) Determine whether A is bounded, and if so what is its diameter.

(b) Determine whether A is closed in $\mathcal { C } _ { \mathrm { b } } ( [ 0 , 1 ] )$

(c) Determine whether A is compact in $\mathcal { C } _ { \mathrm { b } } ( [ 0 , 1 ] )$

(3) Determine the values of $x \in \mathbb { R }$ for which

$$
\sum _ { n = 1 } ^ { \infty } { \frac { x ^ { n } } { 1 + n | x | ^ { n } } }
$$

converges.

(4) Suppose that $f : [ 0 , 1 ] \to \mathbb { R }$ is differentiable and $f ( 0 ) = 0$ . Assume that there is a $k > 0$ such that

$$
| f ^ { \prime } ( x ) | \leq k | f ( x ) |
$$

for all $x \in [ 0 , 1 ]$ . Prove that $f ( x ) = 0$ for all $x \in [ 0 , 1 ]$

(5) Use the Riemann condition to show that $f \in \mathcal { R } _ { \alpha } [ 0 , 4 ]$ where $f ( x ) = e ^ { 2 x }$ and

$$
\alpha ( x ) = \left\{ { x + 1 , \quad 0 \leq x \leq 2 \atop 3 x - 2 , \quad 2 < x \leq 4 } \right.
$$

Compute the value of the Riemann-Stieltjes integral $\int _ { 0 } ^ { 4 } f ( x ) d \alpha .$

(6) Use the Heine-Borel Theorem to prove that if $f$ is continuous on $[ a , b ]$ and $f ( x ) > 0$ for every $x \in [ a , b ]$ then there exists $\varepsilon > 0$ such that $f ( x ) \geq \varepsilon$ for every $x \in [ a , b ]$