Let A = {z ∈ R : there exists n, m ∈ N such that z = 1/n − 1/m }. Prove, using definition(s) and/or result(s) from the lectures, that sup(A) = 1

Question

This is what I have. Please let me know if I am on the wrong track?

Step 1: Let ∈ . Then a = to 1/n - 1/m for some n.

a = 1/n - 1/m < 1/n - 1/m + 1 = 1

so 1 is the upper bound of A.

Step 2:

suppose x < 1, then there is some n such that:

x < 1/n - 1/m < 1

x + 1/m < 1/n < 1+ 1/m

m / xm < n

Since x < 1, there is some n in the natural numbers such than x is not the upper bound of A, hence sup(A) = 1

Is this right?

Answer 1

You write $$a=\frac1n-\frac1m<\frac1n-\frac1m+1=1.$$ Of course the latter equality is false. Fortunately it is also unnecessary. To show that $1$ is an upper bound, note that $$a=\frac1n-\frac1m<\frac1n\leq1.$$ This indeed shows that $\operatorname{sup}A\leq1$. Your proof that equality holds is a bit unclear; indeed for every $x<1$ you want to find $m,n\in\Bbb{N}$ such that $$x<\frac1n-\frac1m.$$ This will show that $x<\operatorname{sup}A$, and hence that $\operatorname{sup}A\geq1$. You proceed with $$x + \frac1m < \frac1n < 1+ \frac1m,$$ which is correct, but then you state that $$\frac{m}{xm}<n,$$ which does not follow; consider $x=\tfrac12$ and $n=1$ and $m=3$. And what if $x=0$?

Instead, note that for all $m,n\in\Bbb{N}$ you have $$\frac1n-\frac1m\leq\frac11-\frac1m=1-\frac1m.$$ With this in mind, it suffices to find some $m\in\Bbb{N}$ such that $$x<1-\frac1m.$$