Problem Set IV

for Thursday, February 16

Due at Adrian Ioana's office hours, or earlier.

Section 2.4, page 51:

• Problem 7.
• Also: Show that Cauchy sequences are bounded (directly from the definition, without assuming the axiom of Cauchy completeness).

Section 2.5, page 54:

• Problem 2.
• Problem 10(a). Suggestion: Take the set of numbers x such that infinitely many terms of the sequence are at least as large as x. Show that the sequence converges to the supremum of this set.
• Also: Show that every real number is the least upper bound of the set of smaller rational numbers. That is,
  x = sup{q | q is rational and q < x}.