Golden-Thompson via pinching inequality

For two hermitian matrices $A$ and $B$, Golden-Thompson inequality \cite{golden1965lower, thompson1965inequality} states that $$\mathrm{tr}\left[ \exp{(A+B)} \right] \leq \mathrm{tr}\left[ \exp{(A)}\exp{(B)} \right].$$ We elaborate here the proof from Sutter et al. (2017).