Algebraic Topology-Explanation required for the following definition I am currently reading the book A combinatorial introduction to topology by Michael Henle.
Under "Compactness and Connectedness" there is the following definition which I didn't understand at all.  I do know the definition of nearness.

Let $P=\{P_1,P_2,P_3,\ldots \}$ be a sequence of points.  The point $Q$ is near the sequence $P=\{P_n\}$ if either $Q=P_n$ for an infinite number of terms of the sequence or $Q=P_n$ only a finite number of times and $Q$ is near the set of other values of $P$.

Any help would be greatly appreciated.
 A: The definition of "$Q$ is near the set $A$" is "every neighborhood of $P$ contains a point of $A$". 
Now, a sequence is really a function $P$ from the set of positive integers to the space. We write $P_n$ instead of $P(n)$, but $P$ is really a function. The author uses set notation, and writes $P=\{P_1,P_2,\dots\}$, but this is not quite correct, since there is a difference between the sequence, which is a function, and its range, which is a set. So, for example, if $P$ is the sequence $P_1=1,P_2=-1,P_3=1,P_4=-1,\dots$ and if $P'$ is the sequence $P'_1=P'_2=1,P'_3=P'_4=P'_5=\dots=-1$, then both are very different as sequences, but have the same range, namely $\{-1,1\}$. 
To say that $Q$ is near a sequence $P$ means that, either for infinitely many values of $n$, $Q$ is the value taken by the function $P$ at $n$, $Q=P_n$, or else this fails, but $Q$ is near the set resulting from excluding $Q$ from the range of $P$. 
To illustrate, for the sequence $P$ given by $P_1=1,P_2=-1,P_3=1,P_4=-1,\dots$ Here, the point $Q=1$ is near $P$, because $Q=P_1=P_3=P_5=\dots$ Also, the point $R=-1$ is near $P$, because $R=P_2=P_4=\dots$ On the other hand, no other point $S$ is near $P$, because if $S\ne1$ and $S\ne-1$, then there is a neighborhood of $S$ so small that neither $1$ nor $-1$ is in it.
Now, look at the sequence $P'$ given by $P'_1=P'_2=1,P'_3=P'_4=P'_5=\dots=-1$. As before, if $S\ne1$ and $S\ne -1$, then $S$ is not near $P'$. Also, $R=-1$ is near $P'$ as $R=P'_3=P'_4=P'_5=\dots$ However, $Q=1$ is not near $P'$, because there are only two values of $n$ for which $Q=P'_n$, namely, $n=1$ or $n=2$. The set of other values of $P'$ is just $-1$, and there is a neighborhood of $Q$ that misses $-1$, so $Q$ is not near the set of other values. 
All this being said, in practice, most sequences $P$ we are interested in will not repeat values, so a point $Q$ will be near $P$ iff $Q$ is not in the range of $P$, but $Q$ is near the range of $P$. For example, $0$ is near the sequence $P$ given by $P_n=1/n$ for $n=1,2,\dots$ 
It is in order to avoid having to phrase arguments awkwardly, and to avoid unnecessarily splitting them into two cases, that the definition is presented to cover both possibilities, when $Q$ is repeatedly listed in the sequence $P$, and when is actually being ``approached'' by the sequence.
