| Problem Set |
Chapter |
Problems |
|---|---|---|
| 03 | 2 | 12 |
| ∗7 | Suppose that a ≤ b. Show that the closed interval [a, b] is compact under our definition of compact set. (Use the definition directly; do not appeal to the Heine-Borel theorem if, indeed, you know what that is.) Hint: Given an open cover {Ωα} of [a, b], define the set E := {x | a ≤ x ≤ b and [a, x] is covered by finitely many Ωα }. Show that sup E is in E, and that sup E = b. | |
| n | manditory problem |
| ∗n | manditory problem, not from text |
| (n) | helper problem |
| [n] | ungraded problem |
| [∗n] | ungraded problem, not from text |
| {n} | optional problem |