Confusion on the definition of accumulation points

A limit point is the same thing as an accumulation point, and its definition is that:

A point $x$ is a limit point of a set $A$ if for every neighbourhood $S$ of $x$ there exists $y \in S$ such that $y \in A$, $y \neq x$.

I strongly prefer the name "accumulation point", because you are not actually taking limits here... it's the other way around! In order to be able to do limits you normally require accumulation points, since the topological definition of a limit requires taking neighbourhoods and computing the function there.

About your second question:

A point $x$ is an accumulation point for a sequence $\{x_n\}$ if any neighbourhood $S$ of $x$ is such that there are infinitely many indices $n$ such that $x_n \in S$.

It is essentially the same definition as above, but you take $A=\{x_n\}_{n \in \mathbb{N}}$. However, a point is a limit point for a sequence if all the indices after a certain $n$ are in any neighbourhood. Formally:

A point $x$ is the limit of a sequence $\{x_n\}$ if any neighbourhood $S$ of $x$ is such that there exists $N \in \mathbb{N}$ such that $x_n \in S$ for all $n>N$.

And this is stronger than simply being an accumulation point: you can see the difference by considering the sequence $x_n = \frac{(-1)^n n}{n+1}$. Any neighbourhood of $1$ contains infinitely many points of this sequence, namely all the $x_{2n}$ after a certain $n$. Similarly, any neighbourhood of $-1$ will contain all the $x_{2n+1}$ after a certain $n$, so both $1$ and $-1$ are cluster points for $x_n$. However, there is no limit (in fact limits are unique, if they exist).


There is a difference between limit and limit point. The concept is defined for sequences and functions but limit point is defined for sets, as mentioned in above answer. A sequence may have limit point but no limit. For example let $\{a_n\}$ is defined as $$1+\frac{1}{n} , (-1)+ \frac{1}{n},... $$ That $a_n=1+\frac{1}{n} $ for odd n's and $a_n=-1+\frac{1}{n} $ for evens. In this sequence both $1$ and $-1$ are limit point but the sequence is not convergent and there is no limit.

Tags:

Limits

Calculus

Sequences And Series

Real Analysis

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