proof verification - Prove that HM of two natural number is not a natural number

I've been trying to prove this

Given any $x,y \in \mathbb{N}$ with $x < y$, $\dfrac{2xy}{x+y} \not\in \mathbb{N}$.

Here's what I tried:

For contradiction, suppose there were some $x,y \in \mathbb{N}$ such that $\dfrac{2xy}{x+y}\in \mathbb{N}$. Let $\dfrac{2xy}{x+y}=H$. Then $y\left( 2\dfrac{x}{H}-1 \right)= x$. Since $H$ is the harmonic mean of $x$ and $y$, we have $0

Is my proof correct? What would be other way to do it?

Answer

Your proof can't be right, since what you're trying to prove is not true.

In particular $(x,y)=(3,6)$ is a counterexample: $\frac{2\cdot 3\cdot 6}{3+6}=\frac{36}{9}=4\in\mathbb N$.

For this counteexample it is true that $\frac xH = \frac 34$ is not a natural number. And $2\frac xH-1 = \frac12$ is not a natural number either. But when we multiply that with $6$, the denominator disappears.