A very quick question on interpreting upper bounds of a union $S\cup T$

Question

Let $S \subset\Bbb{R}$ and $T \subset \Bbb{R}$ be nonempty and bounded. Further suppose that $M$ is an upper-bound of $S \cup T$.

Could someone point out my mistake in the following interpretation please?

$M$ is an upper-bound of $S\cup T$ $\Rightarrow$ $x \leq M $ for all $x \in S \cup T$ $\Rightarrow$ Two possible cases :

[case 1 : $x \in S$]

With $x \in S$ and $x \leq M$, $M$ is an upper bound of $S$ and by the completeness axiom there exists a supremum such that $x \leq \sup(S) \leq M$

[case 2 : $x \in T$]

With $x \in T$ and $x \leq M$, $M$ is an upper bound of $T$ and by the completeness axiom there exists a supremum such that $x \leq \sup(T) \leq M$

Now,intuitively I thought that the following interpretation is correct:

1. For $x \in S \cup T$, either $\sup(S) \leq M$ or $\sup(T) \leq M$

However, when I look around on the internet the results there seem to suggest that:

2. For $x \in S \cup T$, $\sup(S) \leq M$ and $\sup(T) \leq M$.

Continuing from (2) with :

• Therefore $M \geq \max(\sup(S),\sup(T))$
• And finally $M \geq \sup(S \cup T) \geq \max(\sup(S),\sup(T))$

This I am sure is a silly question...but could someone explain why (1.) is wrong and (2.) is correct please?

Answer 1

The two cases you wrote down are not mutually exclusive, and so statement $(1)$ is not true in general (assuming that "either-or" refers to exclusive or; if it refers to inclusive-or, then it is implied by $(2)$).

The correct approach, which leads to $(2)$, is to note that since $M$ upper-bounds $S\cup T$, it upper-bounds every element in $S$ and every element in $T$ at once, and thus $\sup S\le M$ and $\sup T\le M$.

Answer 2

Upperbound $M$ of $S\cup T$ is automatically an upperbound of $S$ and is automatically an upperbound of $T$.

From this we conclude that $\sup S\leq M$ and $\sup T\leq M$.

This tells us that: $$\max(\sup S,\sup T)\leq M$$

From $\sup S\leq M$ and $\sup T\leq M$ it follows directly that also $\sup S\leq M$ or $\sup T\leq M$.