Why is the following statement impossible for a sequence $\{p_n\}$? 
A sequence is said to be convergent if there is a point to which it converges. A convergent sequence cannot converge to two distinct limits. For if $\{p_n\}$ were to converge to both $p$ and $q$ with $p \ne q$, then we could choose spherical neighborhoods about $p$ and $q$ that are disjoint; but, all but a finite number of the $p_n$ would have to be inside each of these neighborhoods, which is impossible.

I'm a little confused by the following statement.

but, all but a finite number of the $p_n$ would have to be inside each of these neighborhoods, which is impossible.

Is it impossible by definition of a convergent sequence?

It is impossible to converge to two different points by definition of convergence described by @copper.hat on the comment above. You cannot have only finite number of values outside two disjoint neighborhoods 
