Five. (20 points) Given the functions
$$
\begin{array}{l}
f(x)=x \ln x, \\
g(x)=-x^{2}+a x-3(a \in \mathbf{R}) .
\end{array}
$$
(1) If for any $x \in(0,+\infty)$, the inequality $f(x) \geqslant \frac{1}{2} g(x)$ always holds, find the range of $a$;
(2) Prove that for any $x \in(0,+\infty)$, we have
$$
\ln x>\frac{1}{\mathrm{e}^{x}}-\frac{2}{\mathrm{e} x} .
$$