Question 1. (a) Suppose that is a set of real numbers that is bounded from below, and let . (By the way, we know that this infimum exists since the set
can be shown to be bounded above and so has a supremum. It is not hard to argue that and so exists.)
Proof. If , then we are done. Therefore, suppose . To show that is an accumulation point, let be given. We will argue that
contains at least one element that is different from . By the Lemma on page 35. (and the one discussed in class), this is equivalent to the original definition of accumulation point.
Since , it follows that for every . By definition of infimum, the real number must fail to be a lower bound for . This means that there must exist an element that satisfies . We therefore have
Since and, by assumption, , it follows that . Since such an element exists for every , we have that is an accumulation point.
(b) There are lots of way to answer this question, but first let’s consider an incorrect answer. The set where with is an open interval. The infimum of this set, and is an accumulation point.
One may think to adjust this example by instead defining the interval so that , however this example still does not work since is still an accumulation point for .
We need to construct a set that (1) has an infimum, (2) contains its infimum, and for which (3) the infimum is not an accumulation point — that is, we want to not accumulate around , so we would like this point to be isolated. Here is one way to take our attempt from before and “fix” it: define the set to be
Then, clearly, but there are -neighborhoods of the point that do not intersect (except in the point itself).
Question 2. There are many ways to prove that this sequence converges. Here is a proof using the definition of convergence.
Claim: The sequence converges to .
Proof. Let be given.
Choose and suppose . Then
where the final inequality is a direct result of our choice for .
Question 3. (a) Claim. The set is bounded under the given (discrete) metric.
Proof. To verify our claim we need to find a point and a non-negative real number so that the set containment
is true. As it turns out, there are infinitely many choices for and infinitely many choices for . In particular, for every and for every , the ball contains every real number. To prove this stronger statement, let and be arbitrary.
Let be arbitrary. To show that we need to show that . However, because and , it follows that . Hence .
As a particular example, we have that . .
(b) Claim. If is a convergent sequence of points in the metric space , then there exists so that whenever it follows that . (In other words: a convergent sequence is eventually constant.)
Proof. Suppose converges to in the given metric space. By definition of convergence this means that for every there exists so that
Since this happens for every positive , it must hold when . As in our statement above, let denote the index beyond which we have
Because this metric can only return values of or , it follows that whenever
Because this is a metric space, the above line implies that whenever , , as desired.
(c ) The Bolzano-Weierstrass Theorem is false in this metric space. As a concrete counter-example, consider the set
Because and is bounded, the set is bounded, too. The set is infinite since there are infinitely many distinct real numbers of the form .
However, does not have any accumulation points. Under our usual metric for , the number would be an accumulation point, but under this discrete metric all of the points of remain of distance away from . Indeed, this is true for all real numbers .
Indeed, one can prove a much more striking fact: in this metric space, no set has any accumulation points. The idea underlying this fact can be stated in a rather simple, English sentence: infinitely many points of any set will fail to accumulate anywhere since they are all distance one away from every other point!
Question 4. Oh, boy. I am skipping this one as I would babble endlessly about failed ideas and how much they (eventually) taught me. Trust me, I’m doing you a favor.
Question 5. Before we address either part of this problem, we establish the following helpful lemma. It will be used in both parts (a) and (b).
Lemma 1. If are subsets of real numbers that each have a supremum, then . (An analogous statement is true for infimuma, but with a reversed inequality.)
proof. Suppose are subsets of real numbers, each with suprema. To prove that , we need only argue that is also an upper bound for the set . Since, by definition, is the least upper bound for , the inequality will have to follow.
Of course, that is an upper bound for is instant. After all, the is an upper bound for and so for all . Now let be arbitrary. Then since , it follows that and so . Therefore is an upper bound for .
(a) To prove that for a bounded sequence of real numbers, , the e-sup always exist we use Lemma 1 to prove that the sequence of suprema
is a decreasing sequence.
By definition we have that
and that .
Using and we have that and so, by Lemma 1,
Similarly, since holds for all , Lemma 1 implies that . Therefore the sequence is decreasing.
Note that we now have an automatic upper bound for , namely for all . To establish a lower bound for we use (again) the fact that the original sequence is bounded. This means that there exist so that
Hence, for every , it follows that since then , too.
The monotone convergence theorem implies that the decreasing, bounded sequence converges. Its limit — which is defined to be the e-sup of — therefore exists.
Note: A very similar proof can be used to prove that the e-inf — which is defined to be the limit of a sequence of infimuma — also exists.
(b) We will handle this iff proof with two different if-then proofs.
proof (of ). Suppose that converges to . We will show that the e-sup = , too. This will be accomplished by proving that for every , it follows that
The first inequality above follows from Theorems in 1.3. In particular, note that by definition of , for each , it follows that . Since both sequences have a limit, theorems from 1.3 imply that .
For the second inequality, let be given. Since, by assumption, , there exists so that whenever , it follows that . Worded differently,
Lemma 1 implies that the supremum of the first set is therefore bounded above by the supremum of the second set. The supremum of the first set is of course , and the supremum for the -ball is . That is, we have
Since is a decreasing sequence we actually have that for all , the suprema . Taking a limit as then produces the desired inequality, namely that
Since is arbitrary, this establishes that . A similar argument can be made to conclude that .
proof (of ). Now suppose that the e-inf and e-sup exist and equal the same number . We will prove that by appealing to our squeeze theorem.
Note, again, that by definition of ,
Similarly, by definition of ,
We therefore have for each ,
Since, by assumption, , the squeeze theorem implies that
Question 6. (Skipped for now — but, yes, these are very similar or outright identical to homework problems, and a proof for part (a) can be used to prove one of the cases you’d naturally set up for part (b).)
Question 7. There are lots of ways to prove this “stronger” Nested Interval Property. My favorite way uses the Monotone Convergence Theorem.
Let be an indexed family of closed, non-empty, bounded intervals that are nested. Observe that we therefore have for all ; observe also that since the intervals are nested it follows that
(We can also note that and that and so on.)
In other words, we have that the sequence is increasing, and the sequence is decreasing. Since both of these sequences are bounded — and — the Monotone Convergence Theorem implies that each sequence converges.
Let us write and , and recall from the proof of the Monotone Convergence Theorem, that these two limits satisfy and . We now treat the following sub-claim:
Sub-Claim: If then .
sub-proof. Suppose but that, say, . Since is the supremum of , this means that there exists a term in the sequence, so that . This inequality implies that
and so as assumed. The case when is handled similarly. .
We will now use the assumption that to prove that . From useful Theorems in sectoin 1.3, we know that since each sequence and converges to and (respectively), the sequence also converges to . However, we also know that
The Nested Interval Property tells us that there exists at least one point , and by our sub-claim this point must lie inbetween and . However, since implies that , there is only one value that can equal. In particular, this all shows
which completes the proof.
Question 8. Let and define . There are lots of ways to argue that . My personal favorite uses an algebraic “trick” that let’s us rewrite the terms as
We can now use various theorems and techniques from Section 1.3 to argue that this sequence converges to zero. For example one can prove (either by induction or more direct methods) that
from which we can conclude that
Some of our results from Section 1.3 then imply that the sequences and both converge to zero. By our Squeeze Theorem, this implies that converges to , too.
We have shown that the sequence of successive distances converges to zero, and this seems like it should imply that the original sequence is Cauchy. However, this is not the case! In particular, that these distances converge to zero does not necessarily tell us anything about the distances between other terms further out in the sequence; that is, we do not necessarily know that
or that, more generally, for any pair ,
Indeed, we can argue that while the sequence converges to zero, the original sequence is not Cauchy. We can accomplish this via a proof by contradiction.
Suppose that is Cauchy. This is means that converges. As a consequence, we know that this sequence must therefore be bounded. That is, there must exist a real number so that
This inequality is equivalent to claiming that there exists so that
Given our familiarity with natural numbers, we can cite a contradiction right here. Therefore, the sequence does not converge and so is not Cauchy.
Note: Some of you very cleverly proved directly that is not Cauchy. If you would like to see one of these proofs, ask around!
Question 9. We will approach this problem by proving two claims. Assume the hypotheses of the question, then
Claim 1. If converges, then it must converge to zero.
Claim 2. The sequence converges.
The second claim is actually the trickier of the two (imo). The first claim can be proven by appealing to our arithmetic laws that govern convergent sequences.
Proof (of claim 1). For a contradiction, suppose that converges to a non-zero number, with . Note further that, by assumption, none of the terms of are equal to zero, and so we may form the sequence
Moreover, by using our results from section 1.3, since converges to , and since also converges to we have that
The claim that only holds if, as we’ve assumed, . This conclusion, however, contradicts the assumption that converges to a number .
Therefore, if converges, it must converge to
There are also many ways to prove claim 2. The proof below will show that the sequence is monotone and bounded, and therefore must converge by the Monotone Convergence Theorem. Actually, we will prove that the sequence is eventually monotone and bounded.
Proof (of claim 2). We are told that the sequence contains only positive numbers, and so we immediately have a lower bound: for all .
Now we will show that this sequence is eventually decreasing. That is, that there exists so that whenever , . The idea for this claim comes from the fact that the sequence converges to , and so the fractions eventually look very close to the number . This implies that, far enough out in the sequence, and so .
To make this idea more precise, we know that for any there exists so that
holds whenever . Since there exists a positive number so that — choose such an , and denote the associated index by . We have that whenever
and this inequality is equivalent to
The upper bound inequality above is what we focus on now. By choice of , we have
Since this last inequality is equivalent to , we have that beyond the index , our remaining sequence is decreasing. Note that this provides for us an upper bound for the sequence , namely the number .
The Monotone Convergence Theorem implies that this sequence converges. This, of course, implies that the original sequence converges, too.
Question 10. (skipped for now)