Question

for set A={1/n:n is positive integer},
why its limit point is {0} ?

Answer 1

Because the limit of its associated function, \(\large f(x)=\frac{1}{x}\), as x approaches infinity is 0.

Answer 2

how to explain if based on the definition of limit point?

Answer 3

Its limit is 0 because for an arbitrary \(\epsilon\) there exists an integer \(n_0\) such that
\[n>n_0 \Rightarrow \left | \frac{1}{n}\right |<\epsilon\].
If I recall correctly, \(\LARGE n_0=\frac{1}{\epsilon}\).