Work as many of the following problems as you can. Be sure your proofs are readable and, for the most part, rigorous. You are encouraged to turn in any positive work on a problem, even if you do not complete all parts. The exam is due by Friday afternoon, May 10.
- (True-False) Decide which of the following statements are true and which are false. Prove the true statements and give counter examples for the false ones.
- If is a convergent sequence and is a divergent sequence, then must diverge.
- If and are both divergent sequences, then must diverge.
- If and are convergent sequences such that for every n, then
- If , then converges.
- If converges and , then converges.
- If converges to , then .
- If and if there exists a sequence converging to a such that , then
- a. Prove: If and , then is eventually less than b. (In other words, there exists such that whenever .)
- Prove: If and , then is eventually greater than a. (If you prove a., then you may say that the proof of b. is similar.)
- Let be a sequence of positive terms such that . Prove the following:
- If , then , (Hint: First show that is eventually deceasing and, therefore, converges (why?); then show that the limit must be 0. Note: Problems 2 and 1. f. above may be useful.)
- If , then . (Hint: Show that is eventually increasing but does not converge.)
- Find a sequence such that and .
Find a second sequence such that and .
- a. Let be a sequence of positive terms. Prove that the infinite series either converges or diverges to . (In other words, assume for every n and let be the sequence of partial sums . Prove that either converges or diverges to .)
- Suppose that and are two positive term series such that for every n.
Prove: i. If diverges, then diverges too.
- If converges, then converges too.
This is known as the comparison test for positive term series.
- Let be an open interval containing a and suppose f and g are two functions defined on .
- Prove: If , then f is bounded on a deleted -neighborhood of . (In other words, show there exists and M such that if then .)
- Prove: If f is continuous at a, then f is bounded on -neighborhood of . (This is almost a direct consequence of part a.)
- Prove: If and g is bounded on a deleted -neighborhood of a, then . (Do not assume exists.)
- Show by counter example that c. is no longer true if .
- Find a continuous function that is a surjection (onto) but does not have a fixed point. (i.e. for every )
- Let be a continuous function such that and . Prove that f has an absolute minimum value on ℝ. (In other words, show there exists ℝ such that for every ℝ .)