Given this partial differential equation:
$$\left\{\begin{matrix} u_t = u_{xx}\\ u(x,0)=g(x), x\in\mathbb{R} \end{matrix}\right.$$
The solution is:
$$u(x,t) = \frac{1}{\sqrt{4\pi t}}\int_{\mathbb{R}}e^{\frac{-(x-y)^2}{4t}}g(y)dy$$
The proposition that we have to prove is:
$$|u(x,t)| \leq \max_{x\in\mathbb{R}}|g(x)| ,\forall t>0$$
Any ideas?