Is the element $0$ in the set $K = \{1/n : n\in \Bbb N\}$?
And moreover, if $0\in K$, can we choose it to do operation like set-minus, such as resuling in
$0 \notin(-1,1)-K $?
2026-04-12 20:53:55.1776027235
Is $0$ in the set $K = \{1/n : n\in \Bbb N\}$?
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As noted in the comments, no, $0 \not \in K$. Were $0 \in K$, then there would exist some $n \in \Bbb N$ such that $1/n = 0$. But multiplying by $n$, a nonzero quantity, on both sides would imply $0=1$, a contradiction. Sure, it is true that
$$\lim_{n \to \infty} \frac 1 n = 0$$
but this is only a "limiting behavior," i.e. it is what the function approaches as $n$ grows larger and larger, and makes no statement as to whether some such element satisfies $1/n = 0$. (Nor does it suggest $\infty \in \Bbb N$. Infinity is an entity entirely separate from the natural numbers.)
...granted, this question is quite old, so I imagine you don't need help now. But hopefully this helps someone in the future, and, if nothing else, gets this question out of the unanswered queue.