= Solution
In the conventions $\{Q_\alpha,Q_\beta^\dagger\}=2\sigma^\mu_{\alpha\dot\beta}P_\mu$, the trace over the two spinor components of the <Super-Poincaré algebra> gives
$$
\boxed{H=\frac14\sum_{\alpha=1}^2\{Q_\alpha,Q_\alpha^\dagger\}}.
$$
Hence, on a normalized state in the domain of the <supercharges>,
$$
\langle v|H|v\rangle
=\frac14\sum_{\alpha=1}^2\left(\|Q_\alpha v\|^2+\|Q_\alpha^\dagger v\|^2\right)\ge0.
$$
This proves <energy positivity in global supersymmetry>. The zero of energy is fixed by the algebra; adding an arbitrary constant to $H$ while keeping that algebra unchanged is not allowed.
For a finite-dimensional <supermultiplet> at fixed positive energy and <four-momentum>, choose a nonzero component $q$ with $\{q,q^\dagger\}=cI$, $c>0$, and $q^2=0$. The Hermitian odd operator
$$
A=\frac{q+q^\dagger}{\sqrt c}
\quad\text{satisfies}\quad A^2=I,
\qquad A(-1)^F=-(-1)^F A.
$$
It is therefore an invertible map from the even <fermion parity> subspace to the odd subspace and conversely. This proves <boson-fermion degeneracy in a supermultiplet>, \b[$n_B=n_F$], for every positive-energy physical particle <supermultiplet>. Equivalently, the <supertrace> of the supercharge <anticommutator> vanishes because odd operators interchange these two subspaces. A zero-energy vacuum is an exception to the literal claim about every possible representation: it can be a one-dimensional bosonic singlet with all <supercharges> acting trivially.
For a vacuum $|\Omega\rangle$, the sum-of-squares formula gives $E_0=0$ exactly when all <supercharges> and their adjoints annihilate it. If global <supersymmetry> is spontaneously broken and a vacuum exists, at least one such norm is nonzero, so \b[the vacuum energy is positive]. In infinite volume this is the statement about the <vacuum energy> density, understood with a finite-volume regulator. Explicit <supersymmetry breaking> need not preserve the algebra used in the proof. The conclusion also does not extend to the <supergravity F-term potential>, whose negative $-3e^K|W|^2$ term permits, for example, the zero-energy broken vacua of <no-scale supergravity>.
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