Fix $b > 1$. If $x$ is real, define $B(x)$ to be the set of all numbers $b^t$, where $t$ is rational and $t \le x$. Prove that $b^r = \sup B(r)$ when $r$ is rational. Hence it makes sense to define $b^x = \sup B(x)$ for every real $x$.
The first part of the question is rather easy and has been proven by an answer to a previous post. Now I am wondering whether the second half, defining $b^x$ for every $x \in \mathbb{R}$, needs to be rigorously proven, and if so, how.
My intuition tells me for each $x$, $b^x$ can be defined as $\sup B(r)$ where $r$ is the greatest rational smaller than $x$. I am unsure about how to communicate this intuition mathematically (given it is correct).