Why dense subsets are convenient to prove theorems Could you please explain the following concept (preferably by examples) about dense subsets:

If you want to prove that every point in $A$ has a certain property that is preserved under limits, then it suffices to prove that every point in a dense subset $B$ of $A$ has that property.

What are the examples of properties that are preserved under limits?
Why is it sufficient to prove such properties for a dense subset $B\subseteq A\subseteq closure(B)$?

Let $B$ be a dense subset of $A$. Suppose property $P$ is preserved under limits and we know that every point in $B$ satisfies property $P$. 
Now pick an arbitrary point of $A$, say $a$. Then $a=\lim_n b_n$ of elements in $B$ each of which have the property and since our property even holds for the limit, it holds for $a$.
Motivation: The best and most relevant examples of this occur when you are working in a space $X$ where your points are functions. So we can talk about several properties like continuous, (insert-mathematician's name)-integrable, differentiable, bounded, harmonic, analytic, etc. 
To give but one important example to highlight the importance of such a notion, let me mention that there have been entire generations of mathematicians whose illustrious careers were dedicated to studying the conditions under which the integral of a limit of functions is the limit of the integral.
