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Final Exam – Math 117, Winter 20201

 

1. (10 points) We have seen in class that every Cauchy sequence of real

numbers is convergent. In this problem we are going to prove this result

in a difffferent way from the way it was done in class. Let {xn}n∈N be a

Cauchy sequence of real numbers:

(a) (5 points) Assume that there exists a convergent subsequence

{xnk }k∈N. Prove that the sequence {xn}n∈N converges.

(b) (5 points) Prove that if {xn} is a Cauchy sequence, there exists

a convergent subsequence. Conclude that every Cauchy sequence

converges.

2. (20 points) The following problems deal with continuous functions:

(a) (5 points) Suppose that f : R → R is continuous and

f(r) = r2

for each rational number r. Determine f(

√ 2) and justify your

conclusion.

(b) (5 points) Let g : [a, b] → [a, b] be a continuous function. Prove

that there exists at least one point c ∈ [a, b] such that g(c) = c.

(c) (5 points) Use the -δ defifinition to show that f(x) = 2x2 + x

1

is continuous at any point in R.

(d) (5 points) Let f : R → R be continuous and let

x

0

∈ R. Assume

that f(x0) > 0. Prove that ∃

δ > 0 such that f

(x

) >

0 for x ∈

(x0

δ, x0 + δ).

3. (20 points) Let {xn}n∈N, {yn}n∈N ⊂ R be bounded sequences.

(a) (5 points) Is it true in general that

lim inf (xn · yn) = (lim inf xn) · (lim inf yn)?

Prove or give a counterexample.

1All course materials (class lectures and discussions, handouts, homework assignments,

examinations, web materials, etc) and the intellectual content of the course itself are

protected by United States Federal Copyright Law, and the California Civil Code. The

UC Policy 102.23 expressly prohibits students (and all other persons) from recording

lectures or discussions and from distributing or selling lecture notes and all other course

materials without the prior written permission of the instructor.

12

(b) (10 points) Assume that {xn}n∈N is convergent and lim xn = L >

0. Prove that

lim inf(xn · yn) ≥ L(lim inf yn).

(c) (5 points) Show that if L ≤ 0, the previous result is not necessarily

true