Math 414: Analysis I
Homework 1
Due: January 27, 2014
Name:
The following problems are for additional practice and are not to be turned in: (All
problems come from Basic Analysis, Lebl )
Exercises: 0.3.1, 0.3.6, 0.3.8, 0.3.10, 0.3.11, 0.3.13, 0.3.14, 0.3.15, 0.3.16
Turn in the following problems.
1. Prove that
∞ \
∞
[
{k ∈ N : mk < n} = ∅
n=1 m=1
yet
∞ [
∞
\
{k ∈ N : mk < n} = N.
m=1 n=1
This shows that we cannot arbitrarily switch unions and intersections.
2. Let f (x) := 1/x2 , x 6= 0, x ∈ R.
(a) Determine the direct image f (E) of E := {x ∈ R : 1 ≤ x ≤ 2}.
(b) Determine the inverse image f −1 (G) of G := {x ∈ R : 1 ≤ x ≤ 4}.
3. Prove that the function f defined by
f (x) := √
x
x2 + 1
(1)
where x ∈ R, is a bijection of R onto {y : −1 < y < 1}.
4. Using induction, prove that
n
X
i=1
1
n
=
,
(3i − 2) (3i + 1)
3n + 1
for n ∈ N.
5. Exercise 0.3.4 from Basic Analysis, Lebl.
6. Prove or give a counterexample to the following statement: If f : R → R and
g : R → R are bijections then f + g is also a bijection.
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Math 414: Analysis I
Homework 1
Due: January 27, 2014
7. (Extra Credit) In class, we used mathematical induction to prove that for all
c 6= 1,
1 + c + c2 + . . . + cn =
1 − cn+1
.
1−c
(2)
Try and prove this result without using mathematical induction.
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