The definition (from my book) we take is:
$x$ is a limit point of $(x_n)$ if $\exists$ a subsequence $(x_{n_k})$ of $(x_n)$ so that $\lim_{k \to \infty} x_{n_k} = x $
Now, this how I understand equivalences to this definition and its negation:
${\bf I.}$ $x$ is limit point of $(x_n)$ if $\forall \epsilon >0$ $\exists\ N > 0$ so that $k > N$ implies $|x_{n_k} – x | < \epsilon $.
${\bf II}.$ If we put put $X = \{ x_n : n \in \mathbb{N} \}$ if for any $\epsilon > 0$, we have infinitely many of the $x_n's$ lying in $B_{\epsilon}(x)$
Now, these are equivalent to the definition of a limit point.
The negation would be: $x$ is ${\bf not}$ a limit point if $\exists\ \epsilon >0$ such that for all $N > 0$ one can find $k > N$ so that $|x_{n_k} – x| \geq \epsilon $
Is my understanding correct?
Best Answer
${\bf II}.$ seems right but ${\bf I}.$ needs to be modified as follows:
${\bf I'}. x $ is a limit point of $(x_n)$ if there exists a sequence $(n_k)_k$ in $\mathbb{N}$ such that $n_k \to \infty$ and for each $\epsilon>0,$ there exists $N \in \mathbb{N}$ such that $k>N$ implies $|x_{n_k}-x|<\epsilon.$
The negation would also need to tweaked as follows:
$x$ is not a limit point of $(x_n$) if for every sequence $(n_k)_k$ in $\mathbb{N}$ with $n_k \to \infty,$ there exists $\epsilon >0$ such that for every $N \in \mathbb{N},$ there exists $k>N$ with $|x_{n_k}-x|\geq \epsilon.$