Proof of Product Law (Limit Laws)

Proving the product law of limits rigorously with the epsilon-delta definition.

REMARKS:
If we have a function h, with domain D, where h(x) = f(x)g(x), we may evaluate its limit by decomposing this function into the product of the functions f and g, where f and g both have domain D, and limits L and M at point c respectively.

The product law tells us that the limit of h(x) (also written as (fg)(x) to emphasize the fact the function can be split into the product of two other functions) is simply LM.

Of course all of this goes without saying that we've assumed point c to be a cluster/accumulation point of the domain D. It makes no sense to talk about the limit if this condition wasn't satisfied.

This law can help us establish a whole range of results without having to go through the hassle of establishing an epsilon-delta definition all the time.