Let $T$ be theory such that there exist model $\mathbb{A}$ for $T$ such that $\mathbb{A}$ is isomorphic with its proper substructure. Prove that there exist infinite model for $T$ which is isomorphic with its substructure.
I have little trouble with that one. Isn't this obvious? Does not from the fact that structure is isomorphic with its proper substructure follows that it is infinite?
Yes, this seems completely trivial: if $\mathbb{A}$ is isomorphic with one of its proper substructures, then $\mathbb{A}$ must be infinite, so $\mathbb{A}$ itself is an infinite model of $T$ which is isomorphic to one of its proper substructures.
I wonder if the problem is asking something slightly different; e.g. "Prove that there exists a countably infinite model of $T$ which is isomorphic with one of its proper substructures." (This is still quite easy, but not completely trivial.) Or "Prove that for every infinite cardinal $\kappa$ there exists a model of $T$ of cardinality $\kappa$ which is isomorphic with one of its proper substructures" - this one I'd say is actually nontrivial (although still not hard).