= Solution
Apply the <Jensen inequality> to the convex <exponential function> under the trial equilibrium measure:
$$
\left\langle e^{-(H-H_0)}\right\rangle_0\geq e^{-\langle H-H_0\rangle_0}.
$$
The given identity therefore implies $e^{-F}\geq e^{-F_0-\langle H-H_0\rangle_0}$. Taking minus the <logarithm> reverses the inequality and proves the <Feynman-Bogoliubov inequality>
$$
\boxed{F\leq F_0+\langle H-H_0\rangle_0=F_0-\langle H_0\rangle_0+\langle H\rangle_0.}
$$
The units $\beta=1$ make $H$ and $F$ dimensionless here. Equality holds when $H-H_0$ is constant on the support of the trial measure. Optimizing the trial parameters tightens an upper bound, which is the basis of the <Gaussian variational approximation>.
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