I've reached a point in a larger proof where I determined that a subset of the rational numbers, $X_n$ is bounded from below. I now need to prove that $X_n$ has an infimum in $\mathbb R$.
Intuitively this is clear, since the infimum would just be min($X_n$) if $X_n$ if it is a closed interval or isolated numbers, and would just be "$a$" if $X_n$ was an open interval ($a$,$b$). However, I am struggling to convert this into an actual proof.
My attempt was to just choose the lower bound L such that $\exists$ L s.t. $\forall$ x $\in$ $X_n$, L $\leq$ x & if $\exists$ L' s.t L'$\leq$ x $\forall$ x $\in$ $X_n$, then L'$\leq$ L, which would make L the infimum, but that seems as though I'm only defining an infimum and not showing that it actually exists.
The statement in your title is equivalent to the axiom of completeness. If you have a definition of the real numbers, for instance as Dedekind cuts of the rationals, then you can prove it.