Question: D = { mn/(1+m+n) } for m,n natural numbers.

To simplify the expression, I presumed m=n, which means:  

D = { n^2 / (2n + 1) }

Now, I know by intuition **there is no supremum**, for this series convergencing to infinity, but I couldn't find a formal way to prove it.

I thought about proving by contradiction:  

- assume S is the supremum  

- ϵ > 0  

S - ϵ > n^2 / (2n + 1) 

But then I got stuck.. 

Any guideness will be very appreciated.  

Thanks!

Answer: 
**Hint**: Try rewriting the fraction by doing division. If you are working in the reals, the least-upper-bound property tells you that any set with an upper bound has a supremum, so if you want to show the set has no supremum, you should be showing it has no upper bound.
**Hint:**
$$\frac{n^2}{2n+1} \ge \frac{n^2}{3n} =n/3$$