MAT2125 Assignment 2
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Assignment 2
MAT2125-A00
Information about assignment 2.
• The deadline of assignment 2 is March 15 11:59pm (Ottawa time).
• Please submit a single pdf file for your solution and name it in the format: Name Student ID Number A2.pdf
• Please include your name and student ID number in your uploaded pdf file.
• You are allowed to collaborate with your classmates in solving the problems.
1. (a) (2 points) Verify the identity
1 1 4
k(k + 2) (k + 2)(k + 4) k(k + 2)(k + 4) ,
(b) (3 points) Show that
∞
X k(k + 2)(k + 4) = 96 .
∀ k ∈ N. (1)
(2)
2. Determine whether or not the following series are convergent. You need to sketch a short proof.
(a) (2 points)
X nn . (3)
(b) (3 points)
X n3 + 1 . (4)
3. (5 points) Consider the Dirichlet function
f : R → R, f(x) =
Show that the Dirichlet function is nowhere continuous.
4. Use the “ϵ − δ” argument to verify the following limits.
(a) (2 points)
lim = 2. (6)
(b) (3 points)
x(l)
x · sin 
= 0. (7)
5. Let f : [0, 1] → R be a continuous function. If f(x) = 0 for any x ∈ Q ∩ [0, 1], show that f ≡ 0 on [0, 1]
Hint. Q ∩ [0, 1] is dense in [0, 1].
6. Let A ⊂ R. A function f : A → R is Lipschitz continuous if there is a positive number
M > 0 such that
f(x) − f(y) ⩽ M x − y , ∀ x,y ∈ A. (8)
(a) (2 points) Show that a Lipschitz function must be uniformly continuous. (b) (3 points) Consider the function
h : [0, 1] → R, h(x) = 
. (9)
Show that h is uniformly continuous on [0, 1] but not Lipschitz.
Hint. The function h is the inverse of the continuous function g : [0, 1] → [0, 1],g(x) = x2 . Take y = 0 in (8) and show that limx→0 h(x)/x = ∞ .
2022-03-16