Advanced Calculus II -- Spring 2017 -- midterm

Name:_________________________ $\qquad$ 7 problems, 70 points total

• Write the problems in the given order, each problem on a separate page.
• Show enough work to justify your answers.

$\bullet$ 1. Suppose $X$ is a set. Prove that the following is a metric: $d(x,x)=0, d(x,y)=1$ when $x\ne y$.

$\bullet$ 2. Suppose $X$ is a metric space. Prove that there are disjoint open balls around $x$ and $y$ whenever $x\ne y$.

$\bullet$ 3. Suppose a function $f:X\to Y$ between two metric spaces is constant: $f(x)=c$ for all $x\in X$ and some $c\in Y$. Prove that $f$ is continuous.

$\bullet$ 4. Does the sequence of functions $f_n(x)=x^n$ converge in $C[0,1]$?

$\bullet$ 5. Prove that every subsequence of a convergent sequence converges.

$\bullet$ 6. Prove that the union of two compact sets is compact.

$\bullet$ 7. Suppose $f:X\to {\bf R}$ is continuous. Prove that $-f$ is also continuous.