I have the following definition of an accumulation point of a sequence:
$P$ is an accumulation point of the sequence $(x_n)^{\infty}_{n=1}$ in $\mathbb{R}$ if: $$\forall \epsilon>0, \ \ \forall N\in\mathbb{N}, \ \exists n\in\mathbb{N} \ \ with \ \ n>N \\ s.t \ \ |x_n-P|<\epsilon$$
I am not sure if this definition is fine. I know the existance of other definitions such as the one given here https://en.wikipedia.org/wiki/Limit_point and the definition given in Terance Tao analysis book (image below). But I would want to know if someone has encounter something like this before.
Thanks for your help.
Best Answer
If $P$ is not in $\lbrace x_n \rbrace$ this definition is equivalent to the one you provided in Wikipedia (I can not see the one in your image) in $\mathbb{R}$.
Assume that $P$ is an acumulation point acording to your definition, then for any neighbourhood $Q$ of $P$ you can define a ball of center $P$ and radius $\epsilon$ small enough such that the ball is inside $Q$. Then by taking $N$ such that $\vert x_n - P \vert$ for all $n \geq N$ you get a point inside $Q$ (for example $X_{N+1}$).
On the other hand if $P$ is an acumulation point acording to the definition in your link then you can construct a sequence $ \lbrace x_n \rbrace$ as the definition that you proposed.
Observe that you did not need to be in $\mathbb{R}$ only to be in a space where the distances are defined.
However if $P$ is in the sequence the definitions are not equivalent, for example look at the case $x_n=P \ \forall \ n$. Then according to your definition $P$ is an accumulation point but it is not for the one you provided in Wikipedia.