if (un) be a monotone bounded sequence, prove that exactly one of l.u.b and g.l.b. of (un) does not belong to (un)

Question

My attempt

Let's assume that the sequence $(u_n)_n$ is both monotone (either increasing or decreasing) and bounded. We need to prove that exactly one of the least upper bound (l.u.b) and greatest lower bound (g.l.b) of the sequence does not belong to the sequence.

First, let's consider the case when the sequence is increasing. Since the sequence is bounded, it has a least upper bound (l.u.b) and a greatest lower bound (g.l.b). We'll show that the l.u.b does not belong to the sequence.

Assume, for contradiction, that the l.u.b belongs to the sequence. Let's denote the l.u.b as $L$. Since $L$ is the l.u.b, it is an upper bound for the sequence $(u_n)_n$. This means that for every term in the sequence, $u_n\leq L$.

However, since the sequence is increasing, we can find a term in the sequence, let's call it $u_{n^*}$, such that $u_{n^*} > L$. This contradicts the assumption that L is an upper bound for the sequence. Therefore, the l.u.b cannot belong to the sequence.

Now let's consider the case when the sequence is decreasing. Similarly, we'll show that the g.l.b does not belong to the sequence.

Assume, for contradiction, that the g.l.b belongs to the sequence. Let's denote the g.l.b as G. Since G is the g.l.b, it is a lower bound for the sequence (un). This means that for every term in the sequence, un ≥ G.

However, since the sequence is decreasing, we can find a term in the sequence, let's call it un*, such that un* < G. This contradicts the assumption that G is a lower bound for the sequence. Therefore, the g.l.b cannot belong to the sequence.

In both cases (increasing and decreasing), we have shown that either the l.u.b or the g.l.b of the sequence does not belong to the sequence. This proves the desired result.

Note that if the sequence is not monotone, then the statement may not hold. In such cases, it is possible for both the l.u.b and g.l.b to belong to the sequence.

Is my proof is write? If wrong, then please prove that question.

Answer 1

Your proof is NOT OK. In the middle of the proof, you write a statement that you did not really justify

However, since the sequence is increasing, we can find a term in the sequence, let's call it $u_{n^*}$, such that $u_{n^*} > L$.

This statement is a unjustified. How do you know this? I mean, in general, it is not true. For example, if $u_n=1-\frac 1n$, and $L=10$, then you cannot find such an $u_{n^*}$. You need to write more clearly how you got $u_{n^*}$ and where it came from.

Furthermore, in your entire proof, you do not really use the fact that $L$ is supposed to be an element of the sequence. Without using that fact, if your proof was valid, it would actually be a proof that a least upper bound of any sequence does not exist. Look, here is my proof of that statement:

Statement: For an increasing sequence, the least upper bound cannot exist.

Proof: Assume for contradiction that the l.u.b. exists. Let's denote the l.u.b as $L$. Since $L$ is the l.u.b, it is an upper bound for the sequence $(u_n)_n$. This means that for every term in the sequence, $u_n\leq L$.

However, since the sequence is increasing, we can find a term in the sequence, let's call it $u_{n^*}$, such that $u_{n^*} > L$. This contradicts the assumption that $L$ is an upper bound for the sequence. Therefore, the l.u.b cannot belong to the sequence.

Do you see the absurdity of my conclusion? Yet I am using the very same justification as you in your proof. This should be a clear red flag to you: if your reasoning can be used to prove a false statement, then the reasoning is flawed.

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In terms of clarity:

First, it's nice to be clear that we are talking about strictly monotone sequences here. The statement clearly is not true for non-strictly monotone sequences.

Second, you use standard mathematical notation which means writing $\inf$ instead of g.l.b. and $\sup$ instead of l.u.b.

Third, you should certainly use mathjax to format your proof. See here for a quick guide: MathJax tutorial . MathJax allows you to be more clear in your writing, as currently, when you write un*, it is not clear if you mean $u_{n^*}$ or $u_n^*$. It also makes everything nicer altogether.