I'm currently learning about supremums but I'm having trouble understanding them.
I understand that for a number M to be the sup(S) it satisfy two conditions:
(1) M is an upper bound of S. (2) If M' is any upper bound of S then $M \leq M'$
The first condition is just saying that for all $ s \in S$, $s \leq M$. Right?
I see why the second condition is true.. But I don't really understand how to use it.
I don't understand how to use it when evaluating problems dealing with supremums. How do we prove that for any upper bound $M'$ of set S, $M \leq M'$.
Also, I'm having trouble with the following example:
Let $a_n$ and $b_n$ be two bounded sequences of real numbers, show that: $$sup(a_n + b_n) \leq sup(a_n) + sup(b_n)$$