Thank you in advance for any more help!

Prove that S = { n−1/n |n ∈ N} is bounded above and that its supremumn

is equal to 1.

(Hint: First show that 1 is an upper bound for S. Then suppose, for a

contradiction, that sup S = α < 1. Using a theorem from the lecture

we can ﬁnd a rational number alpha < p < 1. Deduce that we can ﬁnd an

n ∈ N with αn infiniti alpha< n/n-1 . Contradiction to infiniti alpha being an upper bound.)