Additivity of suprema

Question

I'm currently starting with a more formal approach of calculus. I am reading Apostol's Calculus Vol. 1 and found the proof of the additivity of suprema confusing. Could someone please do it and explain it without skipping any steps.

The conditions given in the book are:

Let A and B be nonempty sets that have a supremum. Given that $$ C=\{a+b | a\in A, b\in B\} $$ Prove that: $$ \sup C=\sup A +\sup B $$

Thanks in advance for the help!!!

Answer 1

If $a\in A$ and $b\in B$, then $a+b\leqslant\sup A+\sup B$ and therefore $\sup A+\sup B$ is an upper bound of $C$. So, $\sup C\leqslant\sup A+\sup B$.

Now, suppose that $\sup C<\sup A+\sup B$. Take $a\in A$ and $b\in B$ such that $\sup C<a+b\leqslant\sup A+\sup B$. But $a+b\in C$ and so we can't have $a+b>\sup C$.

Can you finish the proof by proving that if $\sup C<\sup A+\sup B$, then there must be an $a\in A$ and a $b\in B$ such that $\sup C<a+b$?