How do I find the supremum and infimum of set A?
$$A = \{ \frac{5^{3n}*5^{2m}}{3*5^{5n}+5^n + 2*5^{5m}} , n, m \in \mathbb{N}\}$$
I know that this expression is always greater than 0 since it is made of natural numbers only.
- for $n = 1$, $m \implies \infty$, $A \implies 0$
- for $m = 1$, $n = \implies \infty$, $A \implies 0$
Therefore, I know that the infimum is equal to $0$. However, how do I find the supremum?
From a graph I made I know that for $n = m = 1$ that expression is equal to $\frac{1}{5}$ and that might be the supremum. However, how do I deduct that from this expression? I tried dividing nominator and denominator by $5^n$ and $5^m$. I tried comparing nominator to denominator. Got nothing.
Any hint would be much appreciated.
Hint : Let $x=5^n$ and $y=5^m$. Notice that
$$ \frac{1}{5}-\frac{x^3y^2}{3x^5+x+2y^5} = \frac{1}{5(3x^5+x+2y^5)}\bigg( x+(y-x)^2\big(3x^3 + 6yx^2 + 4y^2x + 2y^3\big)\bigg) $$