How do I explain why the set of natural numbers (N) cannot be equivalent to one of its finite subsets?

You need to carefully understand what equivalent means - that there is a 1-1 correspondence between the sets.

I think the best place to start is the Pigeon Hole Principle: if n+1 pigeons fly into n pigeon holes then at least one pigeon hole contains two pigeons. How does this help with your question if the finite subset has n elements?

Penny