in trying to prove that a function on a closed bounded interval attains a max value.

i was reading a proof and the way they proved it was that to show that

let M be the sup of f then show that

f(x) is less than M-1/M_1 hence M-1/M_1 is an upper bound for f, which is smaller than M..hence M is the max..

but im wondering, if that works, then M-1/M_1 should be the max of f right? and not M..