Supremum & Infimum Homework Statement

[drawar]

Homework Statement

Let $$S = \left\{ {\frac{n}{{n + m}}:n,m \in N} \right\}$$. Prove that sup S =1 and inf S = 0

Homework Equations

The Attempt at a Solution

So I was given the fact that for an upper bound u to become the supremum of a set S, for every ε>0 there is $$x \in S$$ such that x>u-ε. In this case, I'm supposed to find n and m such that $${\frac{n}{{n + m}} > 1 - \varepsilon }$$ for every ε given. However, I cannot express n and m in terms of ε explicitly. Any hints or comments will be very appreciated, thanks!

 

[tiny-tim]

hi drawar!

I'm supposed to find n and m such that $${\frac{n}{{n + m}} > 1 - \varepsilon }$$ for every ε given.

hint: n/(n+m) = 1 - m/(n+m)

 

[drawar]

hi drawar! hint: n/(n+m) = 1 - m/(n+m)

Hi tiny-tim, thanks for the hint. Do you mean:

$${1 - \varepsilon < \frac{n}{{n + m}} = 1 - \frac{m}{{n + m}}}$$
Choosing m=1:
$${\varepsilon > \frac{m}{{n + m}} > \frac{1}{{n + 1}}}$$
and then solve for n?

 

[tiny-tim]

yup!

except, that's $\frac{1}{\frac{n}{m}+1}$ ​