## 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?

**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).