The sequence {n /(n+1)} has a least upper bound of 1 Show that if M is a number less than 1, the

step 1 So we want to show that the least upper bound of n over n plus 1 is 1. So before we start this,

The sequence {n /(n+1)} has a least upper bound of 1 Show...

Key Concepts

Archimedean Property

The Archimedean property states that for any real number, however large, there exists a natural number greater than it. This property is fundamental when working with sequences and inequalities, as it guarantees that one can always find a natural number that surpasses any given bound. It is used in proofs that require selecting an index beyond which all sequence terms satisfy a certain inequality, hence linking boundedness and convergence.

Epsilon (or Margin) Arguments

Epsilon arguments are a formal method used in calculus and analysis to rigorously prove that a sequence converges to its limit. By showing that for every positive number (epsilon) there exists an index after which all terms of the sequence lie within an epsilon distance from the limit, one can validate that the sequence eventually exceeds any number less than its limit. This approach is widely used to demonstrate the precise behavior of sequences near their limits.

Supremum (Least Upper Bound)

The supremum of a set, also known as the least upper bound, is the smallest number that is greater than or equal to every element in the set. In real analysis, identifying the supremum is a fundamental concept because it ensures that even if a set does not contain a maximum element, there is a well-defined bound that no term in the set exceeds. This concept is essential when studying the completeness property of the real numbers.

Convergence of a Sequence

Understanding the behavior of sequences as their index becomes large is crucial in real analysis. A sequence converges if its terms approach a specific value, called the limit, as the index tends to infinity. This concept not only helps in defining limits but also in establishing how closely the sequence's terms can get to a certain bound, which is important when demonstrating properties related to supremum.

Transcript

00:01 So we want to show that the least upper bound of n over n plus 1 is 1.

00:07 So before we start this, let's just go ahead and maybe do some scratch work to kind of give us some intuition as to what we want to do.

00:16 So scratch.

00:20 So we want some little n such that if m is between 0 and 1, then...

00:45 N over n plus 1 is going to be strictly larger than m.

00:54 So now let's just go ahead and do a little bit of math and maybe try to solve this in terms of n on one side as opposed to having these two ends.

01:02 So first thing i'm going to do is reciprocate each side.

01:06 So that would give me n plus 1 over n and 1 over m on this side.

01:11 And remember when i reciprocate like this, i need to switch my inequality.

01:15 Now let's go ahead and divide this n into here.

01:19 So we'd have 1 plus 1 over n.

01:23 Now i'm going to subtract 1 over, and i'm going to have 1 over n is less than 1 over m minus 1.

01:31 And doing that algebra over here, we will go ahead and get 1 minus m over m.

01:38 And now let's reciprocate one more time, and we'll get little n should be larger than m over 1 minus.

01:46 All right, so if we have little n bigger than m over 1 minus n, then we have a pretty good idea that this value should, oh, so this value here should be strictly larger than m.

02:08 All right, so this is just our scratch work, but remember, we're going to want to use this inequality in the next thought of our proof.

02:16 So let's just go ahead and start our proof.

02:18 So proof.

02:22 Now, something we might want to show is that this function is actually bounded above by one first.

02:30 So consider that the limit as an approach is infinity of n over n plus 1 is going to be just 1.

02:43 And you can show this however you want, but we at least know this.

02:50 And also consider that if we take the derivative with respect to n of n over n plus 1, so you could use quotient rule to find this, but i'm just going to go ahead and say what this derivative should be...

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