Questions related to problems from Polya-Szego's book

Question

In the last few days I solved some problems from Polya-Sego's book but right now I think that the formulation of some problems is slightly confusing and nonstandard. That is why I am opening this post and want to clarify some moments.

Question 1.

Suppose that $A_n$ are partial sums of series $\sum \limits_{n=1}^{\infty}a_n$ and $l=\liminf \limits_{n\to \infty}A_n, \ L=\limsup \limits_{n\to \infty}A_n$.

Then we have to show $\{A_n:n\in \mathbb{N}\}$ is everywhere dense in $[l,L]$, right? But usually when we trying to prove that $A$ is dense in $X$ we mean that $A\subset X$, right?

But here some elements of $\{A_n: n\in \mathbb{N}\}$ may be outside of $[l,L]$. And it seems to me a bit confusing.

Question 2.

The phrasing of this question is vague.

1. Is the limit point of a sequence same as limit point of set? In other words, $p$ is a limit point of a sequence $\{t_n\}_{n=1}^{\infty}$ if for any $\epsilon>0$ one can find $t_N$ such that $0<|t_N-p|<\epsilon$, right?

2. If we take $\nu_n=n$ then $\dfrac{\nu_n}{n+\nu_n}=\frac{1}{2}$ and we see that the limit point of that sequence is empty.

So I would be very happy if someone can clarify my thoughts, please.

Answer 1

1. "A is dense in X" means that $\overline{A} = X$. The closure of A is X. An equivalent statement is "for all $\epsilon > 0$, and $x \in X$, there exists a $a \in A$ such that $|x - a| < \epsilon$". Here, a depends on $\epsilon$ and $x$.

2. How many $A_{n}$ can lies outside of $[l,L]$?

3. $x$ is a limit point of the set A if for every neighborhood of x contains an ement of A that is not x. Notice how a limit point is defined for a set as opposed to a sequence. It make sense to talk about cluster points of sequences. The set ${1}$ has no limit point in $\mathbb{R}$ but the sequence $(1/2)_{n \geq 1}$ has a cluster point $1$.

Answer 2

Question 1. A set is everywhere dense on an interval if and only if every subinterval of the original interval contains a point of the given set. So the set can have elements outside of the interval.

Question 2.

1. The limit points of the set of the numbers of the sequence is necessarily the limit points of the sequence. But the converse is not generally true because the sequence may repeat infinitely many times a same number.

2. $\frac12$ is the limit point of this sequence. But the limit point set of its image set is empty, as you say.