Homework 7

Limits

1. The lim sup of a sequence is the largest limit of any subsequence, or $\pm\infty$. What is the lim sup of the following sequences?

(1)
\begin{eqnarray} a_n&=&(-1)^n \\ b_n&=&cos(n) \\ c_n&=&1,-1,2,-2,3,-3,\cdots \\ d_n&=&-n^2\\ e_n&=&\frac12,1+\frac12,2+\frac12,\frac14,1+\frac14,2+\frac14,\frac18,1+\frac18,2+\frac18,\ldots\\ f_n&=&\mathbb{Q}\cap(0,1) \end{eqnarray}

2. Complete problem 5 in Chapter 2 of the text (proof). You may use the fact that the complement of an open set is closed.

Infinite Intersections of Open/Closed Sets

3. Give an example of open sets $U_1, U_2, U_3, \ldots$ such that $U_1\supset U_2\supset U_3 \cdots$ but the intersection $\bigcap_{i=1}^\infty U_i = U_1\cap U_2\cap \cdots$ is closed and nonempty.

4. Complete problem 27 in Chapter 2 of the text (proof).

Inheritance Properties

5. If $S=[0,1)$, what are $\partial S, \overset\circ{S}, \bar S$ (the boundary, interior, and closure)?

6. Prove that the interior of any subset S of a metric space is the largest open set contained in S. That is, show that every open set $U\subset S$ is contained in the interior of S ($U\subset\overset\circ{S}$).

Compactness

7. By completeness, a nonempty compact set of real numbers has both a largest element (maximum) and smallest element (minimum). We proved this in the first block. Find a nonempty set of real numbers which does have a maximum and minimum but is not compact.

8. Prove that the intersection of two compact sets is compact. Likewise, the union of two compact sets is compact. You should use the subsequence definition of compactness in your proof.

Boundedness

9. Find an example of a sequence $(a_n)$ such that (i) every convergent subsequence converges to 0 and (ii) $a_n$ does not converge to 0. (Hint: this is only possible if the sequence is not bounded.)

10. Complete problem 29 in Chapter 2 of the text (proof).