\( \def\naturals{\mathbb{N}} \def\integers{\mathbb{Z}} \def\rationals{\mathbb{Q}} \def\reals{\mathbb{R}} \def\complex{\mathbb{C}} \)
Topics: Compactness, Sequence and Series
Book Chapters: Rudin, Chapter
Due date: Before class, 14 Oct 2015
Mode of submissions: Please submit at blackboard.
Format: Please keep it in pdf. format with your name and andrew id included in it. The filename must be LastName_FirstName-yourAndrewID-HW4.
Expected time required for this homework: Six hours. If you're quick, you can do it in under an hour.

I. Compact and Connected sets

  1. Is the segment (0,1) compact in $\mathbb{R}$? If yes, prove it, otherwise, give an example of an open cover which has no finite subcover in this case.
  2. Is the closed interval [a, b] connected in $\mathbb{R}? a,b \in \mathbb{R}$ and $a < b$. Prove it.

II. Sequence

  1. Prove that a monotonic sequence converges if and only if it is bounded.
  2. Prove that every cauchy sequence converges in $R^{k}$
  3. Prove that every convergent sequence in a metric space is Cauchy.
  4. A sequence $\{x_n\}$ converges to a value $p$. Prove that every subsequence of $\{x_n\}$ also converges to the same value.

III. Sequence

  1. Prove that the series $\sum \frac{1}{n}$ diverges.
  2. Prove that a series of non-negative terms converges if and only if its partial sums form a bounded sequence.
  3. If $\sum a_n$ converges and if $\{b_n\}$ is monotonic and bounded, prove that $\sum a_n * b_n $ converges.