For my calculus class we were asked to show that if $$ f:\mathbb R^{2}\to\mathbb R $$ is of class $C^2$, then the hessian matrix of $f$, $ Hf(x) $, is the symmetric $n\times n$ matrix of second-order partial derivatives, i.e. $${[Hf(x)]}^T=Hf(x).$$ For the proof, I noted that $$Hf(x)_{ij}=\partial_{ij}f(x).$$ Furthermore, $${[Hf(x)]}^{T}_{ij}=\partial_{ji}f(x).$$ Since $f$ is $C^2$ $$\partial_{ij}f(x)=\partial_{ji}f(x),$$ so the proof is complete.
Is my proof correct? It seems overly-simple, so I am not sure.