Pointwise limit of $f$

Question

If we let $f_n(x)= 1,$ if $x\in\{1,\frac{1}{2},...,\frac{1}{n}\}$ and $0$, otherwise. How can I find the pointwise limit $f$ of this sequence?

Answer 1

Argue by cases.

If $x$ is of the form $\frac1k$ for some $k$, then convince yourself that $x\in\{1,\frac12,\ldots,\frac1n\}$ for all sufficiently large $n$, which means $f_n(x)=1$ for all sufficiently large $n$. Thus $f_n(x)\to1$.

If $x$ is not of the form $\frac1k$ for any $k$, then argue that $x$ is never a member of the set $\{1,\frac12,\ldots,\frac1n\}$ and therefore $f_n(x)=0$ for all $n$. Thus $f_n(x)\to0$.