We show that the composition of two isomorphisms is again an isomorphism.
We show that the image of an Abelian group under a homomorphism is itself an Abelian group. That is, homomorphisms preserve commutativity.
We show that congruence modulo n is a congruence relation on the set of integers with respect to both addition and multiplication. We also find the kernel of this relation.
We begin by formally defining a limit. We then show, step-by-step, how to prove that the limit of a function at a given point is equal to a given value.
We show, step-by-step, how to show that a mapping is a bijection. We do this both directly, and by finding an inverse function.