Let Г be a sentence of predicate logic such that for any natural number m ≥ 1, there is a model of Г with at least n elements. Then Г has a model with infinitely many elements.
A model is basically a set. The theorem says that if you have a model larger than any model of finite size, you have a model of infinite size. But that is obvious. It is obvious that if, no matter how larger is your finite model, I can always come up with a larger one, this means that my set contains an infinite model. Even the proof of the principle involves expanding Г with $I_1, I_2, \ldots, I_m$ (where $I_m$ says that model size is larger than $m$), applying Complactness theorem to it (which says that such model, larger than $m$, exists) and finally, my common sense is used: the author says that the model must be infinite because it is larger than any finite m. But is clear from the very beginning. What is the point of introducing all these $I_m$s and applying the compactness?