I am currently reviewing some of the proofs in Spivak's Calculus, and I have come across this statement as a part of the proof for Extreme value theorem.
if $f(x)$ is continuous on $[a, b]$ then for every $\epsilon > 0$ there is $x$ in $[a, b]$ with $L - f(x) < \epsilon$ where $L$ is the least upper bound.
Now this basically mean that its a one sided limit where $f(x)$ approaches $L$ but I wonder why there were no justifications for this step in the book, usually nothing is taken for granted in Spivak's proofs but this seems to be something non-trivial by first year analysis standard that is stated as true. So is this some how "obvious" or is Spivak asking the reader to justify this step instead?
Edit: There were no mentioning of any real number properties what-so-ever. This proof is on page 136 of the third edition.