I'm asked to find all the convergent sequences and their limits on $X$ equipped with the discrete topology. My answer is as follows:
The discrete topology is $\mathcal P \left({X}\right)$. If some sequence $\{x_n\}$ converges to $x$, then all neighborhoods of $x$ intersect $\{x_n\}$ at some point other than $x$. Since the singleton $\{x\}$ is a neighborhood of $x$ that cannot intersect $\{x_n\}$ at any point other than $x$, then $x$ is not a limit point of any sequence in $X$.
However the correct answer is:
Under the discrete topology a sequence $\{x_n\} ⊆ X$ is convergent, only if $x_n$ stabilizes for $n$ sufficiently big, i.e. there exists an $N$ s.t. for all $n ≥ N$, $x_n = x$ for some $x∈X$; in this case $x_n →x$.
So why is my topological definition of convergence failing me?
Best Answer
A sequence $(x_n)_{n\in\mathbb N}$ converges to $x$ when every neighborhhod of $x$ contains every $x_n$ when $n$ is large enough. So, consider the neighborhood $\{x\}$. It contains every $x_n$ when $n$ is large enough. In other words, if $n$ is large enough, then $x_n=x$.