Baby Rudin - Existence of logarithm

I'm working out of Baby Rudin, chapter 1 problem 7. The problem is "Fix b >1, y > 0, and prove that there is a unique real x such that $\displaystyle b^x = y $ by completing the following outline..."

I'm at part f) "Let A be the set of all w such that $\displaystyle b^w < y $, and show that $\displaystyle x = sup(A) $ satisfies $\displaystyle b^x = y $.

What I'm wondering about - if m is a positive integer, since b > 1, is it a leap to assume that there's an integer k such that $\displaystyle b^k > m $?