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 intereseted 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.

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.