Dear users of mathstackexchange, I'm an upcoming 12th grade high school student who wants to self-study Real Analysis. I'll be posting my solutions to every exercises I do from Carothers' Real Analysis textbook in hope of getting constructive feedback.
1.1 If $A$ is a nonempty subset of $\mathbb{R}$ that is bounded below, show that A has a greatest lower bound. That is, show that there is a number $m \in\mathbb{R}$ satisfying: (i) $m$ is a lower bound for $A$; and (ii) if $x$ is a lower bound for $A$, then $x < m$
proof.
Consider the set $L:=\{l\in\mathbb{R}: l \le a\in A\}$. Since $A$ is bounded below, $L$ is nonempty. Choose any $a\in A$. Clearly, $a$ is an upper bound of $L$. Least upper bound axiom gaurantees that $L$ has a supremum, which is enough to imply the existence of the least upper bound of $A$.
This set is undefined, since you haven't clearly defined $a$. I suppose at best you could say that this is a set that depends on $a$, i.e. call it $L_a$.
But I think you're trying to write "the set of lower bounds of $A$," which this is certainly not. You want $$ L := \{ \ell \in \mathbb{R} \mid \ell \le a \text{ for every } a \in A \} $$ Your original writing only specifies a single (heretofore undefined) $a$, whereas this makes it clear that it needs to hold for every $a$.
Maybe so, but you certainly haven't given a convincing argument of such. Why should it be true that $\sup L$ existing implies $\inf A$ exists?
A key issue, in no small part, is you have not clearly shown the existence of an $m$ which satisfies both (i) & (ii) as given. There are plenty of choices that satisfy (i), but you should be able to find something that satisfies (ii) and cleanly demonstrate it does.
(In fact, questions of greater importance: what is the relationship of $\sup L$ and $\inf A$? And how do you prove said relationship?)