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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.
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(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 ﻿