Homework 3 Calculus 1 1. Rudin Chapter 3, Exercise 3. 2. Rudin Chapter 3, Exercise 5. For simplification, assume that sequences {

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Homework 3 Calculus 1

1. Rudin Chapter 3, Exercise 3.

2. Rudin Chapter 3, Exercise 5. For simplification, assume that sequences {a_n} and {b_n} are both bounded.

3. Given a sequence {a_n} of real numbers and define b_k= sup

n≥k

a_n.

Prove that (a)

lim

k→∞b_k= inf

k b_k. (b)

lim

k→∞b_k = lim sup

n

a_n.

Similar arguments show that (you don’t have to do it)

lim inf

n a_n = lim

k→∞inf

n≥ka_n = sup

k→∞

inf

n≥ka_n.

4. Given a nonempty subset E ⊂ R that is bounded above (so that sup E exists), prove that there exists a sequence {a_n} ⊂ E converging to sup E.

5. Given two sequences {an} and {bn} of real numbers with an ≤ bn. Prove that (a) lim sup_na_n≤ lim sup_nb_n.

(b) lim inf_na_n≤ lim inf_nb_n.

(c) If both sequences are convergent, then lim_na_n≤ lim_nb_n.

Also, give counterexample to the statement lim sup_na_n ≤ lim inf_nb_n. 6. If a_n > 0, and lim_n→∞^an+1_a

n = L > 0, then lim_n→∞√ⁿ

a_n = L. (Note that the statement is also true for L = 0).

7. Find the limit of (a) √ⁿ

n.

(b) √ⁿ

n⁵+ n⁴. (c) ⁿ

qn!

nⁿ.

Hint: Use problem 6.

8. Rudin Chapter 3, Exercise 4.

9. Salas 11.4: 7, 20, 23, 24.

10. Salas 12.2: 3, 7, 9, 14, 29.

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