1. (6 pts) True/False. Indicate whether the statement is True or False. (no explanation required)

________(a)Between any two irrational numbers, there is a rational number.

________ (b) Let x be a real number. If7 - e£ x £ 7 for all e > 0, then x = 7.

________(c) If any sequence (xn) diverges, then the sequence (1/xn) converges to 0.

2. (6 pts) Let A = [2, 5]and B = [-1, 4].Define function f: A ® Bby .

 

True or False? (No explanation required.)

________(a)f is an injective function.

________(b)fis a surjective function.

________(c)Let T = {0, -1}. Then pre-image f -1(T) = {2, 5/2}.

3. (8 pts) Prove by induction:for all n Î N.

 

4. (14 pts) True or False? For each part: If true, prove it carefully. You can apply or cite any of our relevant definitions, theorems, examples, or exercises in a proof. If false, provide a counterexample and explain it.

(a)If (an) is any increasing sequence of negative real numbers and (bn) is any Cauchy sequence of real numbers, then the sequence (an - bn) converges.

(b)If{An : n N} is any collection of subsets ofR, with each set An containing only finitely many numbers, then the union is closed and bounded.

 

5. (15 pts) Complete the following table. For each set, determine the infimum, minimum, maximum, and supremum (if they exist). (If there is not a finite value or it does not exist, record an X instead). Indicate whether the set is open,closed, neither, or both.(explanation optional)

Set	Infimum	Minimum	Maximum	Supremum	Open / closed/ neither / both

 

6. (12 pts) For each of the following sequences (sn), state whether the sequence converges or diverges. If the sequence converges, determine the limit. If the sequence diverges to ¥ or to –¥ state the infinite limit. Show work / explanation. You may cite textbook/notes theorems, examples, and/or homework textbook exercises if relevant (by source and page numbers).

(a)

(b)

(c)

7. (3 pts) For each of the sequences in the previous problem, state the subsequential limits (which could be infinite).

Sequence part in #6	Subsequential Limits
(a)
(b)
(c)

 

8. (8 pts) Use the e definition (page 2 of the Week 3 Sequences document or page 44 of Lebl) to prove that

.Tip to consider: Is bounded?

 

9. (7 pts) Consider the following claim and proof.

 

Claim: For all real numbers p and q, if p is an odd integer, then p - 2q is an odd integer.

 

Proof:

p = 2k + 1 by definition of oddinteger.

p - 2q = (2k + 1) - 2q

= 2(k - q) + 1,sop - 2q is an odd integer.

 

 

(a) Critique the proof.Is it complete? Does it prove the claim? What are the flaws, if any? Be specific.

(b) Is the Claim true or false? Explain carefully if the Claim is false. (no additional explanation if true.)

 

 

10. (7 pts) Consider the following claim and proof.

 

Claim: Let S be a subset of R.If the set S has at least one interior point, then the set S is uncountable.

 

Proof:

Suppose S is a subset of R having at least one interior point. Let x denote an interior point of S.

By definition of interior point, there is a neighborhood W of x such that W is a subset of S.

By definition of neighborhood, W is a bounded open interval of real numbers.

The interval W has the same cardinality as the interval (0, 1), by Week 2 Homework, #14(b).

Since the interval (0, 1) is uncountable, W is uncountable, and since W is a subset of S, the set S must also be uncountable.

(a) Critique the proof.Is it complete? Does it prove the claim? What are the flaws, if any? Be specific.

(b) Is the Claim true or false? Explain carefully if the Claim is false. (no additional explanation if true.)

 

 

11. (7 pts) Consider the following claim and proof.

Claim: The sequence given by converges to 0.

Proof:

Let The sequence of ratiosconverges to .

so by the Ratio test for sequences (Week 3 sequence notes, page 9, or Lemma 2.2.12(iii) Lebl),

the sequence converges to 0.

 

(a) Is the Claim true or false? Explain carefully if the Claim is false. (no additional explanation if true.)

(b) Critique the proof.Is it complete? Does it prove the claim? What are the flaws, if any? Be specific.

 

12. (7 pts) Consider the following claim and proof.

 

Claim: Let D be a nonempty compact subset of R, with function f: D ® R.

If T is a nonempty subset ofD and the maximum of D is 2, then sup f (T) £f (2).

Proof:

Let D be a nonempty compact subset of R, with function f: D ® R.

Suppose thatT ¹ Æ, T Í D , and the maximum of D is 2.

Then f (T) ¹ Æ,f (T) Íf (D), and for all x Î T, we have f (x) £ f (2).

So, f (T) is bounded above by f (2).

Thus sup f (T) exists (by the Completeness Axiom), and we must have sup f (T) £f (2).

 

(a) Is the Claim true or false? Explain carefully if the Claim is false. (no additional explanation if true.)

(b) Critique the proof.Is it complete? Does it prove the claim? What are the flaws, if any? Be specific.

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