Your error is in asserting that n-1< x. The Archimedean property only tells you that thereisan integer larger than x. it doesnottell you that that integer is anywherenearx. For example, if the x= 1.5, the Archimedean property might be telling you that the number 1,000,000 is larger than 1.5.

You will also need to use the "well ordering property" of the positive integers: every non-empty set of positive integers contains a smallest member. Apply that to the set of all integers larger than x which the Archimedean property tells you is non-empty.