I know how to prove it by the fact they don't respect the usual change of coordinates, but I want to prove it using that a tensor must be $\mathcal{C}^\infty$-multilinear in all its components. Here is my wrong proof, someone can help me finding the mistake?
Call $\Gamma:=\Gamma_{ij}^kdx^idx^j\frac{\partial}{\partial x^k}$, then I want to prove that $\Gamma$ is not $\mathcal{C}^\infty$-multilinear, i.e. if $X,Y\in\mathcal{T}(M)$, $w\in\mathcal{T}^*(M)$ and $f,g,h\in\mathcal{C}^\infty(M)$, then it doesn't hold that $\Gamma(fX,gY,hw)=fgh\Gamma(X,Y,w)$.
But we have:
$\Gamma(fX,gY,hw)=\Gamma_{ij}^kdx^i(fX^s\frac{\partial}{\partial x^s})dx^j(gY^m\frac{\partial}{\partial x^m})\frac{\partial}{\partial x^k}(hw_tdx^t)=fgh\Gamma_{ij}^kdx^i(X^s\frac{\partial}{\partial x^s})dx^j(Y^m\frac{\partial}{\partial x^m})\frac{\partial}{\partial x^k}(w_tdx^t)=fgh\Gamma(X,Y,X)$
So where's the mistake? Thank you all.
I found it. I can't write $\Gamma:=\Gamma_{ij}^kdx^idx^j\frac{\partial}{\partial x^k}$ since I am assuming that Christoffel symbols are components of the tensor $\Gamma$.