- 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 =
lim
_{n→∞}34∑_{k=1}^{n}^{k}now use Problem 7. - Let a ∈. Show a = .a
_{1}a_{2}a_{3}for some choice of integers in, a_{1},a_{2},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
.010010001000010000010000001. 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 2q
^{2}= p^{2}and so p^{2}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 x
_{1}= b > 0 where b^{2}> a. Explain why there exists such a number, b. Now having defined x_{n}, define x_{n+1}≡. Verify thatis a decreasing sequence and that it satisfies x_{n}^{2}≥ a for all n and is therefore, bounded below. Explain why lim_{n→∞}x_{n}exists. If x is this limit, show that x^{2}= 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 x_{n}, just a rule which tells how to define x_{n+1}if x_{n}is known. - Let a
_{1}= 0 and suppose that a_{n+1}=. Write a_{2},a_{3},a_{4}. Now prove that for all n, it follows that a_{n}≤+. 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 lim
_{n→∞}a_{n}= a, does it follow that lim_{n→∞}=? Prove or else give a counter example. - Show the following converge to 0.
- Suppose lim
_{n→∞}x_{n}= x. Show that then lim_{n→∞}∑_{k=1}^{n}x_{k}= x. Give an example where lim_{n→∞}x_{n}does not exist but lim_{n→∞}∑_{k=1}^{n}x_{k}does. - Suppose r ∈. Show that lim
_{n→∞}r^{n}= 0. Hint: Use the binomial theorem. r =where δ > 0. Therefore, r^{n}=<, etc. - Prove lim
_{n→∞}= 1 . Hint: Let e_{n}≡− 1 so that^{n}= n. Now observe that e_{n}> 0 and use the binomial theorem to conclude 1 + ne_{n}+e_{n}^{2}≤ n. This nice approach to establishing this limit using only elementary algebra is in Rudin [32]. - Find lim
_{n→∞}^{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 [10]. - Find limsup
_{n→∞}^{n}and liminf_{n→∞}^{n}. Explain your conclusions. - Give a careful proof of Theorem 4.9.18.
- Let λ = limsup
_{n→∞}a_{n}. Show there exists a subsequence,such thatNow consider the set S of all points in

such that for s ∈ S, some subsequence ofconverges to s. Show that S has a largest point and this point is limsup_{n→∞}a_{n}. - Let ⊆ ℝ and suppose it is bounded above. Let
Show that for each n, sup

≤ sup. Next explain why sup≤ limsup_{n→∞}a_{k}. 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→∞}a_{n}. Show there exists a subsequence,such that lim_{k→∞}a_{nk}= λ. Now consider the set, S of all points insuch that for s ∈ S, some subsequence ofconverges to s. Show that S has a smallest point and this point is liminf_{n→∞}a_{n}. Formulate a similar conclusion to Problem 19 in terms of liminf and a sequence which is bounded below. - Prove that if a
_{n}≤ b_{n}for all n sufficiently large that - Prove that
- Prove that if a ≥ 0, then
while if a < 0,

- Prove that if lim
_{n→∞}b_{n}= b, thenConjecture and prove a similar result for liminf .

- Give conditions under which the following inequalities hold.
- 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 S
_{n}such that S_{n}⊋ S_{n+1}whose intersection has more than one point. Next give an example of a nested sequence of nonempty sets S_{n}, S_{n}⊋ S_{n+1}which has 2 points in their intersection. - For F = ℝ or ℂ, suppose F = ∪
_{n=1}^{∞}H_{n}where each H_{n}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.

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