Upper Bound Proof of Sup(SUT)=max{sup(S), sup(T)}

[STEMucator]

Homework Statement

Prove or disapprove, for non-empty, bounded sets S and T in ℝ :

sup(SUT) = max{sup(S), sup(T)}

Homework Equations

The least upper bound axiom of course.

The Attempt at a Solution

Since we know S and T are non-empty and bounded in the reals, each of them contains a supremum by the least upper bound axiom. Let : L₁ = sup(S) ^ L₂ = sup(T) be these least upper bounds for S and T respectively.

Since SUT is also a bounded non-empty set, it also contains a supremum by the axiom. Let L = sup(SUT) denote SUT's least upper bound.

We want to show that L = max{L₁, L₂}

Not quite sure how to proceed from here.

 

[voko]

Prove that it can't be less and can't be greater.

 

[STEMucator]

Prove that it can't be less and can't be greater.

Uhm, well since S is a subset of SUT and T is a subset of SUT, we know that SUT which is comprised of all of the points of S and T must also contain the least upper bounds of both sets. That is L₁, L₂ are in SUT.

Is this the right direction?

 

[voko]

This is not true. The least upper bound need not be contained in its set. For example, (0, 1) does not contain its least upper bound, nor does (2, 3), nor will (0, 1) U (2, 3).

 

[STEMucator]

This is not true. The least upper bound need not be contained in its set. For example, (0, 1) does not contain its least upper bound, nor does (2, 3), nor will (0, 1) U (2, 3).

Ah yes I see, so from what you told me before, I must somehow show :

L₁, L₂ < L < L₁, L₂ ?

 

[STEMucator]

So for L to be the sup(SUT), it must be an upper bound, that is x ≤ L for all x in SUT.

It also has to be the least upper bound, that is for any upper bound of SUT, say M, L ≤ M.

Are these points relevant? If so I believe I know how to do this.

 

[jbunniii]

So for L to be the sup(SUT), it must be an upper bound, that is x ≤ L for all x in SUT.

It also has to be the least upper bound, that is for any upper bound of SUT, say M, L ≤ M.

Are these points relevant? If so I believe I know how to do this.

Well, that's the definition of the least upper bound, so it's certainly relevant.

Using your notation, you need to show that both of the following are impossible:

$$L < \max(L_1, L_2)$$
$$L > \max(L_1, L_2)$$

So start by assuming one of these and finding a contradiction.

 

[voko]

Well, these are the definition of the least upper bound. Yes, these could be use to prove that sup SUT is not less and is not greater than max {sup S, sup T}.

 

[jbunniii]

Hint: consider what the two inequalities mean.

$L < \max(L_1, L_2)$ means that either $L < L_1$ or $L < L_2$ (or both).

$L > \max(L_1, L_2)$ means that $L > L_1$ and $L > L_2$

 

[STEMucator]

So I'll write out my whole proof below here :

Since we know S and T are non-empty and bounded in the reals, each of them contains a supremum by the least upper bound axiom. Let sup(S) ^ sup(T) be these least upper bounds for S and T respectively.

Since SUT is also a bounded non-empty set, it also contains a supremum by the axiom. Let sup(SUT) denote SUT's least upper bound.

We want to show that sup(SUT) = max{sup(S), sup(T)}. So to do this we want to show sup(SUT) ≤ max{sup(S), sup(T)} and sup(SUT) ≥ max{sup(S), sup(T)}.

To show sup(SUT) ≤ max{sup(S), sup(T)} consider the following :

Suppose x is in S, then we know x ≤ sup(S) ≤ max{sup(S), sup(T)}.
Suppose y is in T, then we know y ≤ sup(T) ≤ max{sup(S), sup(T)}.

Putting these together we know that sup(SUT) ≤ max{sup(S), sup(T)}.

To show sup(SUT) ≥ max{sup(S), sup(T)} let M be an upper bound for SUT. Then consider that since M is an upper bound for the union of S and T, this tells us that either M is an upper bound for S or M is an upper bound for T.

If M is an upper bound for S, then M ≥ sup(S). If M is an upper bound for T, then M ≥ sup(T). So it follows now that sup(SUT) ≥ max{sup(S), sup(T)}.

Now since sup(SUT) ≤ max{sup(S), sup(T)} and sup(SUT) ≥ max{sup(S), sup(T)}, we only have one option left, it must be that sup(SUT) = max{sup(S), sup(T)} as desired.

 

[voko]

To show sup(SUT) ≥ max{sup(S), sup(T)} let M be an upper bound for SUT. Then consider that since M is an upper bound for the union of S and T, this tells us that either M is an upper bound for S or M is an upper bound for T.

I think it would be more straightforward to say that for any x (or y) from S(or T), x (y) is no greater than M; and because sup S and sup T are the least upper bounds, they are also necessarily no greater than M. Essentially, just like the first part.

 

[STEMucator]

I think it would be more straightforward to say that for any x (or y) from S(or T), x (y) is no greater than M; and because sup S and sup T are the least upper bounds, they are also necessarily no greater than M. Essentially, just like the first part.

Yes I see what you're saying.

Thanks for the help guys :)