- Suppose x = .3434343434 where the bar over the last 34 signifies that this
repeats forever. In elementary school you were probably given the following
procedure for finding the number x as a quotient of integers. First multiply by
100 to get 100x = 34.34343434 and then subtract to get 99x = 34. From this
you conclude that x = 34∕99. Fully justify this procedure. Hint: .34343434 =
k now use Problem 7.
- Let a ∈. Show
a = .a1a2a3
for some choice of integers in ,
if it is possible to do this. Give an example where there may be more
than one way to do this.
- Show every rational number between 0 and 1 has a decimal expansion which either
repeats or terminates.
- Using Corollary 3.2.5, show that there exists a one to one and onto map θ from the
natural numbers ℕ onto ℚ, the rational number. Denoting the resulting countable
set of numbers as the sequence
, show that if x is any real number, there
exists a subsequence from this sequence which converges to that number.
- Consider the number whose decimal expansion is
. Show this is an irrational number.
- Prove is irrational.
Hint: Suppose =
p∕q where p,q are positive integers
and the fraction is in lowest terms. Then 2q2 = p2 and so p2 is even. Explain why
p = 2r so p must be even. Next argue q must be even.
- Show that between any two integers there exists an irrational number. Next show
that between any two numbers there exists an irrational number. You can use the
fact that is irrational if you like.
- Let a be a positive number and let x1 = b > 0 where b2 > a. Explain why there
exists such a number, b. Now having defined xn, define xn+1 ≡
Verify that is a decreasing sequence and that it satisfies
xn2 ≥ a for all n
and is therefore, bounded below. Explain why limn→∞xn exists. If x is this limit,
show that x2 = a. Explain how this shows that every positive real number has a
square root. This is an example of a recursively defined sequence. Note this does
not give a formula for xn, just a rule which tells how to define xn+1 if xn is known.
- Let a1 = 0 and suppose that an+1 =
. Write a2,a3,a4. Now prove that for
all n, it follows that an ≤ +
. Find the limit of the sequence. Hint: You
should prove these things by induction. Finally, to find the limit, let n →∞ in
both sides and argue that the limit a, must satisfy a =
- If limn→∞an = a, does it follow that limn→∞ =
? Prove or else give a
- Show the following converge to 0.
- Suppose limn→∞xn = x. Show that then limn→∞
k=1nxk = x. Give an example
where limn→∞xn does not exist but limn→∞
- Suppose r ∈
. Show that limn→∞rn = 0. Hint: Use the binomial theorem.
r = where
δ > 0. Therefore, rn =
- Prove limn→∞ = 1
. Hint: Let en ≡
− 1 so that
n = n. Now observe
that en > 0 and use the binomial theorem to conclude 1 + nen +
en2 ≤ n. This
nice approach to establishing this limit using only elementary algebra is in Rudin
- Find limn→∞
1∕n for x ≥ 0. There are two cases here, x ≤ 1 and x > 1. Show
that if x > 1, the limit is x while if x ≤ 1 the limit equals 1. Hint: Use the argument of
Problem 14. This interesting example is in .
- Find limsupn→∞
n and liminf n→∞
n. Explain your conclusions.
- Give a careful proof of Theorem 4.9.18.
- Let λ = limsupn→∞an. Show there exists a subsequence, such that
Now consider the set S of all points in such that for
s ∈ S, some subsequence
of converges to
s. Show that S has a largest point and this point is
⊆ ℝ and suppose it is bounded above. Let
Show that for each n, sup
. Next explain why
≤ limsupn→∞ak. Next explain why the two numbers are actually equal.
Explain why such a sequence has a convergent subsequence. For the last part, see
Problem 18 above.
- Let λ = liminf n→∞an. Show there exists a subsequence, such that
k→∞ank = λ. Now consider the set, S of all points in such that for
s ∈ S,
some subsequence of converges to
s. Show that S has a smallest point and this
point is liminf n→∞an. Formulate a similar conclusion to Problem 19 in terms of liminf
and a sequence which is bounded below.
- Prove that if an ≤ bn for all n sufficiently large that
- Prove that
- Prove that if a ≥ 0, then
while if a < 0,
- Prove that if limn→∞bn = b, then
Conjecture and prove a similar result for liminf .
- Give conditions under which the following inequalities hold. Hint: You need to consider whether the right hand sides make sense. Thus you can’t
consider −∞ + ∞.
- Give an example of a nested sequence of nonempty sets whose diameters converge to 0
which have no point in their intersection.
- Give an example of a nested sequence of nonempty sets Sn such that Sn ⊋ Sn+1 whose
intersection has more than one point. Next give an example of a nested sequence of
nonempty sets Sn, Sn ⊋ Sn+1which has 2 points in their intersection.
- For F = ℝ or ℂ, suppose F = ∪n=1∞Hn where each Hn is closed. Show that at least one
of these must have nonempty interior. That is, one of them contains an open ball. You
can use Theorem 4.9.23 if you like.