Solution
= Solution
Let $\psi_1$ be the normalized ground state for
$$
H_1=T+V_1.
$$
Using it as a trial state for $H_2=T+V_2$, the variational principle gives
$$
E_2\leq\langle\psi_1,H_2\psi_1\rangle
=E_1+\langle\psi_1,(V_2-V_1)\psi_1\rangle.
$$
The pointwise assumption $V_2\leq V_1$ makes the last expectation nonpositive. Hence
$$
\boxed{E_2\leq E_1.}
$$