Department of Mathematics

Midterm # 1, MATH-UA.0325 - Fall 2024

Exercise 1. (10 pts) True or false, prove or find a counterexample.

a) Let {xn}∞ n=1 ⊂ R be a bounded sequence. Then, it is not possible to find a convergent subsequence {xnk }∞ k=1 ⊂ {xn}∞ n=1.

b) Assume the real number x > −1. Then,

(1 + x) n ≥ 1 + nx, n > 1.

Exercise 2. (10 pts) Answer to the following questions:

a) Prove that if f maps E → F and A ⊂ E, B ⊂ E, then

f(A ∪ B) = f(A) ∪ f(B).

b) Consider the sets V = {(x, y) : √x2 + y2 < 1} and W = {(x, y) : max{|x|, |y|} < 1}. Prove that V ⊂ W.

Exercise 3. (10 pts) Let

xn = (1 + n2/n2) cos 3/2nπ.

Find lim inf xn and lim sup xn.

       n→∞             n→∞

Exercise 4. (10 pts) Use the Cauchy’s criterion or the ratio test to determine whether or not the following sequences converge (justify your answer):

a) {xn}∞ n=1, where xn = n!3n/n n.

b) {yn}∞ n=1, where yn = 2/sin 1 + 2 2/sin 2 + · · · + 2n/sin n.

Exercise 5. (10 pts) Determine whether or not the sequence {xn}∞ n=1, where

xn = nn/n! ,

converges by answering to the following questions:

a) Is {xn}∞ n=1 monotone decreasing?

b) Is {xn}∞ n=1 bounded from below?

c) Compute lim xn/xn+1.

                n→∞

In addition,

d) Is lim √xn = √lim xn? Justify your answer.

      n→∞          n→∞