This is probably wrong but feels right to me :\
Consider some set $S ⊂ \mathbb{R}$, and say I've found an upper bound $b$ of $S$, such that $\forall s\in S, s \leq b$. This implies that $\sup S \leq b$. Then I showed that this lower bound is actually achievable, i.e., $\exists s^* \in S = b$. Would this then imply $\sup S \geq b$, and therefore $\sup S = b$?
The definition of $\sup S$ is the following : $b = \sup S$ if the following two things happen:
1) $b \geq s$ for all $s \in S$.
2) If any $c$ also has the above property, then $c \geq b$.
You say you have found an upper bound, namely $b$, so the first condition is satisfied. Now, you claim that for some $s^* \in S$, $s^* = b$.
From this how do we prove that $\sup S = b$? Go by definition. Suppose that $c$ has the property that $c \geq s$ for all $s \in S$. In particular, $c \geq s^*$, since $s^* \in S$. However, $s^* = b$, so $c \geq b$ is true. Hence, the second condition is also satisfied.
Hence, $\sup S =b$ is true. I know this proof was a bit long winded, but I want to get beginners into the habit of proceeding by definition, rather than making leaps of logic early on , and deriving even simple propositions by hand. Hence, this answer. Hope it helps.
