Using R3824: Bias-variance partition of mean square error, we have \begin{equation} \mathbb{E} |X - \lambda|^2 = \text{Var} X + (\mathbb{E} X - \lambda)^2 \end{equation} The term $\text{Var} X$ is a constant, so to minimize the right-hand side it is sufficient to minimize the expression $(\mathbb{E} X - \lambda)^2$. The function \begin{equation} \lambda \to (\mathbb{E} X - \lambda)^2 \end{equation} is nonnegative and attains (the minimum) value $0$ at $\lambda = \mathbb{E} X$. Hence \begin{equation} \begin{split} \mathbb{E} |X - \lambda|^2 & = \text{Var} X + (\mathbb{E} X - \lambda)^2 \\ & \geq \text{Var} X \\ & = \mathbb{E} |X - \mathbb{E} X|^2 \\ \end{split} \end{equation} $\square$