# Basic Analysis - Proof Bolzano Wierestrass by Least Upper Bound

• rakalakalili
In summary, the given problem deals with showing the existence of a subsequence that converges to the supremum of a bounded set. By defining s as the supremum of the set S, we can use the fact that for any ε > 0 there exists some x ∈ S with s − ε < x. By choosing εk = 1/k, we can construct a subsequence (a_nk) that converges to s.
rakalakalili

## Homework Statement

Let (an) be a boundedd sequence, and define the set
$S= {x\in R : x < a_n$ for infinitely many terms $a_n\}$
Show that there exists a subsequence $(a_n_k)$converging to s = sup S

## Homework Equations

This is supposed to be a direct proof of BW using the LUB property, so no monotonic convergence, Cauchy criterion, nested interval property etc...

## The Attempt at a Solution

I am having trouble thinking of a way to define the subsequence. What I can show is that, if $\epsilon > 0$ there are finitely many terms $a_n$ s.t. $a_n> s+\epsilon$
I thought of defining the subsequence to be $a_n_k = min(a_n | a_n > s+\frac{1}{k})$ But I was having trouble proving that this subsequence converges to s. I would greatly appreciate a tip or prod in the right direction of defining a subsequence that will work. Thank you!

Note that the set S is bounded above, since (a_n) is bounded and any upper bound of
the sequence will be an upper bound of S. Therefore, sup S exists, define s = sup S.
Now, consider an arbitrary ε > 0. Since s + ε cannot be an element of the set S, there must be only a finite number of an such that an > s + ε. But by the definition of the supremum, for any ε > 0 there exists some x ∈ S with s − ε < x. Thus, for any ε > 0 there are an infinite number of terms an with s − ε < an and only a finite number of those terms satisfy an > s + ε. Therefore, we must have an infinite number of an with s−ε<an <s+ε.
Now, to construct a subsequence (a_nk ) → s, consider εk = 1/k. So, start by choosing some an1 so that s−1 < an1 < s+1, from this point on, we choose ank+1 so that nk+1 >nk ands−1/k<ank+1 <s+1/k.We know that we can always do this, since at every step k there are an infinite number of a_n with s−1/k < a_n <s+1/k.

That's perfect thank you! I had known that there was an infinite number of terms less than s+ \epsilon, but I did not connect that with making an interval, and simply choosing a term from each interval to form the subsequence. Thanks again.

## 1. What is the Bolzano-Weierstrass Theorem?

The Bolzano-Weierstrass Theorem, also known as the Least Upper Bound Theorem, states that any bounded sequence of real numbers has a convergent subsequence. This means that if a sequence of numbers is limited in some way, there will always be a point where the sequence “converges”, or reaches a limit.

## 2. How does the Bolzano-Weierstrass Theorem relate to basic analysis?

The Bolzano-Weierstrass Theorem is a fundamental concept in basic analysis. It is often used to prove the convergence of sequences and functions, which is an important part of understanding the behavior of mathematical functions and their limits.

## 3. What is the significance of the Least Upper Bound in the Bolzano-Weierstrass Theorem?

The Least Upper Bound (LUB) is the smallest number that is greater than or equal to all the numbers in a given set. In the context of the Bolzano-Weierstrass Theorem, the LUB plays a crucial role in proving the existence of a convergent subsequence. It acts as a “limit” for the sequence and ensures that the subsequence does not “escape” to infinity.

## 4. How is the Bolzano-Weierstrass Theorem used in real-world applications?

The Bolzano-Weierstrass Theorem has numerous applications in real-world scenarios, especially in fields such as physics, engineering, and economics. It is used to prove the convergence of numerical methods and algorithms, which are essential in solving complex problems in these fields.

## 5. What are the differences between the Bolzano-Weierstrass Theorem and the Heine-Borel Theorem?

While both the Bolzano-Weierstrass Theorem and the Heine-Borel Theorem deal with bounded sequences, they have different applications. The Bolzano-Weierstrass Theorem focuses on the convergence of sequences, while the Heine-Borel Theorem deals with the compactness of a set. Additionally, the Bolzano-Weierstrass Theorem only applies to real numbers, while the Heine-Borel Theorem applies to any metric space.

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