Proof of theorem in infimum and supremum

Question

The first line is the statement that I want to prove. Let A and B be bounded non-empty subsets of R. Can someone please tell me does my proof (especially second last line) of this question valid or not?

Answer 1

Your proof is ok. Nice job.

Now, hints for improving:

• It is good to point out that you can suppose WLOG that $\min\{\inf A, \inf B\} = \inf A$.
• The part "let $\forall \ c \in A \cup B$" has this unnecessary $\forall$. You can just say "let $c \in A \cup B$" and go on.
• Write more. Don't be afraid to show your ideas explictly.

And since you didn't seem so sure of the second last line, I'll give a proof of what you used.

Lemma: Let $X \subseteq Y \subseteq \Bbb R$ be lower bounded sets. Then $\inf Y \leq \inf X$.

Proof: Take $x \in X$. Since $X \subseteq Y$, $x \in Y$ and so $\inf Y \leq x$. Then $\inf Y$ is a lower bound for $X$, and by definition of infimum, we obtain $\inf Y \leq \inf X$.