This lemma is simple, but it is very useful in rigorous proofs of Statistical Mechanics.

For example the existence of the thermondynamic limit of the free energy per particule $\frac{f_N}{N}$ of an Ising spin model can be proven in many cases by proving the free energy $f_N$ is sub/super-additive. If interested you can see:

*F. Guerra, F. Toninelli - The Thermodynamic Limit in Mean Field Spin Glass Models* .

If someone know any generalisation of this lemma, I'm very interested too.