Question

1/n +1/m n,m€NFind range with explanation.​

Answer 1

Step-by-step explanation:

If S = {1/n − 1/m : n, m ∈ N}, find inf S and sup S.

Answer. We claim inf S = −1. Let 1/n − 1/m be an arbitrary element in S. Then,

1/n − 1/m ≥ 1/n − 1 > −1. So −1 is a lower bound for S.

Let ² > 0. By Corollary 2.4.5, there exists n0 ∈ N such that 1/n0 < ². Now,

1/n0 − 1 < ² − 1 = −1 + ² and 1/n0 − 1 ∈ S. Thus, −1 = inf S.

We claim sup S = 1. Note that S = −S and apply homework problem #5 from

section 2.3 in which you proved inf S = − sup{−s : s ∈ S}. We get −1 = inf S =

− sup{−s : s ∈ S} = − sup S which implies sup S = 1.

is it right ?