Harmonic function and harmonic conjugate

Let $u:Gsubsetmathbb{R} rightarrow mathbb{R}$ a harmonic function $v:Grightarrow mathbb{R}$ the harmonic conjugate function, with $G$ a domain. Prove that $u^2-v^2$ and $uv$ are harmonic without derivatives.

Before this, I proved that $u^2$ is harmonic, if $u$ is an harmonic function. Then, I thought that $u^2$ and $v^2$ are harmonic functions, and I wanted to conclude that $u^2-v^2$ is a harmonic function.

Nonetheless, this interpretation is wrong.

Thanks in advance