Firstly, the set E is defined: "Let E be the set of all positive real numbers t such that tn<x." Later on the proof goes: "Assume yn>x. Put k=yn-x / nyn-1. Then 0 < k <y. If t ≥ y - k, we conclude that yn-tn ≤ yn-(y-k)n < knyn-1 = yn-x. Thus tn>x, and t is not a member of E. It follows that y - k is an upper bound of E." Why does it follow? Is it because the possibility that t = y - k combined with the fact that tn>x mean that y-k always has to be an upper bound of E? Or is there some other reasoning? Thanks in advance.