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Modern Analysis
Homework 3. Due: Friday, September 16, 2011, 10AM
Note. None of these exercises is terribly difficult. In fact,
some are fairly trivial, others quite easy. Their main pointis to
see if you understand the definition of limit and if you can write
mathematics without making unwarrantedassumptions, inventing new
strange notation, using undefined concepts, etc. Can you stick to
the basics, in otherwords. Can you try to first understand, then
write? Will you read what you write and see if it makes sense to
you?
1. The reason for this exercise: In my opinion, most of you
missed the point of Exercise 1 of Homework 2.
Explain what is wrong with the following proof: I will
prove:
sin x + cos x
tan2 x 1=
cos2 x
sin x cos x
for all x for which the expressions are defined. Multiply both
sides by sin x cos x, get
sin2 x cos2 x
tan2 x 1= cos2 x.
Now multiply both sides by tan2 x 1, get
sin2 x cos2 x = cos2 x(tan2 x 1).
Using that tan2 x 1 =sin2 x
cos2 x 1, we can multiply out the right hand side of the last
equality to get
sin2 x cos2 x = sin2 x cos2 x,
which is obviously true. The proof is complete.
2. The definition of limn an = L given in the text is: For every
> 0 there is N N such that |an L| < whenever n > N.
Explain why the following two definitions are equivalent:
(a) For every > 0 there is N N such that |an L| < whenever
n N.
(b) For every > 0 there is N R such that |an L| < whenever
n > N.
3. Prove: Let {an} be a sequence of real numbers. Then limn an =
0 if and only if limn |an| = 0.
4. What is the negation of the statement: limn an = L? That is,
negate the definition:
For every > 0 there exists N N such that n > N implies |an
L| < .
Or, if you do not like the word implies, negate the definition
in the form
For every > 0 there exists N N such that |an L| < whenever
n > N.
The negation is NOT limn an = L.
5. Let {an} be a sequence of integers. Can it converge?
Discuss.
6. Here is another equivalent way of stating that a sequence of
real numbers converges. The book calls it anunofficial definition.
Let {an} be a sequence of real numbers. Prove: It converges to the
number L if andonly if for each positive real number the set {n N :
|an L| } is finite.
I phrased it in a slightly different (contrapositive) way from
what Professor Zorn calls the unofficial definition(Exercise 13 of
2.1). His statement is: if and only if for every > 0 , we have
|an L| < for all but finitelymany n. Spend perhaps a few minutes
convincing yourselves that my version and his version are
equivalent.In proving the equivalence keep in mind:
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You are proving an if and only if statement, so you have to go
in two directions.
Both statements, the original definition and its if and only if
consequence, begin with for every, orfor each, > 0. So maybe
both parts of the proof might begin with Assume > 0 is
given.
In one direction you have a finite set and have to produce N N.
In the other direction you have N Nand have to produce a finite
set. What could be the relation between a natural number and a
finite set?
7. Prove the following sequences converge. Youll have to guess
the limit.
(a)
3n2
(2n + 1)2
.
(b){
2n}
.
(c){
2nn3}
.
8. Let A be a non-empty set of real numbers, assume A is bounded
above and = sup A. Prove: There existsa sequence {an} of points of
A (an A for all n N) such that limn an = .
9. Let an be defined, for n N by
an = 2n if n is even,1
n if n is odd.Prove using the definition that limn an = 0.
