= Solution
Let $\{\phi_j\}$ be a complete <orthonormal eigenbasis> of the <Hamiltonian>, with $H\phi_j=E_j\phi_j$ and $E_j\geq E_0$. Expand a nonzero admissible <trial wavefunction> as $\widetilde\Psi=\sum_j c_j\phi_j$. Then
$$
\langle\widetilde\Psi|\widetilde\Psi\rangle=\sum_j|c_j|^2,
\qquad
\langle\widetilde\Psi|H|\widetilde\Psi\rangle=\sum_j|c_j|^2E_j.
$$
Consequently
$$
\widetilde E-E_0
=\frac{\sum_j|c_j|^2(E_j-E_0)}{\sum_j|c_j|^2}\geq0,
\qquad\boxed{\widetilde E\geq E_0.}
$$
Every numerator term is nonnegative and the denominator is positive. Equality requires the trial to lie wholly in the <ground-state subspace>; it need not be an individually chosen eigenvector when the ground <energy> is degenerate. This is the <quantum variational principle>, and <wavefunction normalization> before calculation is optional because the quotient removes the overall scale.
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