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→∞