# 代写MATH数学作业 MATH 142B Assignment 2B

## MATH 142B Assignment 2B

Only complete this assignment if you have been assigned to Group B.
If you have been assigned to Group A, complete Assignment 2A instead.

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Only complete this assignment if you have been assigned to Group B.
If you have been assigned to Group A, complete Assignment 2A instead.

1. Abbott Exercise 2.6.5.

2. Begin by reading the statement and proof of Theorem 2.3.4 in Lebl’s textbook.

Suppose {xn} is a convergent sequence, {yn} is a bounded sequence, and xn ≥ 0, yn ≥ 0 for every n ∈ N. Set x = lim xn and y∗ = lim sup yn.

(a) Use the theorem mentioned above to show that {xnyn} admits a subsequence converging to xy∗.
(b) Use a result from HW1A to deduce that lim sup(xnyn) = xy∗

Remark. The theorem referenced in this problem gives an alternate proof to the Bolzano-Weierstrass theorem than that given than that presented in Fitzpatrick’s textbook (which is probably the textbook you used for MATH 142A if you are in Group B).

### 3. Suppose {xn} is a sequence with xn 6= 0 for each n ∈ N, and set  代写MATH数学作业 Suppose L < 1.
(a) Show that if L < a < 1, then there exists N ∈ N so that if n ≥ N, then (b) Deduce that Pxn converges.
(c) Prove that P converges.
4. (a) Abbott Exercise 2.7.8(a).
(b) Abbott Exercise 2.7.8(b).
(c) Abbott Exercise 2.7.8(c). 