Why does unbroken supersymmetry imply the absence of tachyons?

Just a quick question, same as in the title. I'm trying to understand stable D-branes.

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In theories with unbroken supersymmetry, the energy can be written as $$E=\sum_i c_i Q_i Q_i$$ where $Q_i$ are some Hermitian supercharges and the coefficients are positive. This is a sum of squares of Hermitian operators which is why it's positively semidefinite. It can't be negative.
Tachyons obey $E^2-p^2=m^2$ for a negative $m^2$ so the 4-momentum (or $d$-momentum) is a spacelike vector, assuming it is real. For spacelike vectors, one may always change $E$ to a negative number by a Lorentz transformation, but no such states with $E\lt 0$ may exist due to the positivity above, which means tachyons can't exist at all.