We have two themes this week: the first one is the topology of metric space side. We discussed the notion of compact set. I recommend Rudin's book ch2. There are two major theorems, the Heine-Borel theorem, which applies to $\R^n$; and the compactness = sequential compactness, which is true for general metric space. See Rudin Theorem 2.41 (page 40)

The second theme is series. It is related to sequence by considering partial sum, and we reuses many of results from sequence. The alternating series test and the integral test was new, but one should have met them in calculus.

Rudin Ch2 page 43-45

1. Rudin Ex 2.11.

2. Rudin Ex 2.14.

3. Rudin Ex 2.15

4. Ross 14.14

5. If $a_n > 0$ and $\sum_n a_n$ diverges, show that $\sum_n a_n / (1 + a_n)$ diverges.

6. If $a_n > 0$ and $\sum_n a_n$ converges, show that $\sum_n \sqrt{a_n}/n$ converges.

Hint: Cauchy inequality: for any real valued $N$-tuples $(A_1, \cdots, A_N)$ and $(B_1, \cdots, B_N)$, we have $$ (\sum_{j=1}^N A_j B_j)^2 \leq (\sum_{j=1}^N A_j^2) (\sum_{j=1}^N B_j^2)$$