Suppose that for every positive integer $n$, we have a function $f_n(x)$ such that $∀ x ∈ {1, 1/2, . . . , 1/n}, f_n(x) = x$ and at all other real numbers $x, f_n(x) = 0$. Find the pointwise limit function $f$ of $(f_n)$ on $R$.
As $n$ tends to infinity, the interval on which $f_n(x)$ is non-zero becomes smaller and smaller (as $1/n$ $\rightarrow$ $0$). So does this mean that the point-wise limit is $0$?
Hint
Let $ q $ be an integer $\ge 1 $. Then
$$n\ge q \implies \frac 1q\in\{1,\frac 12,\cdots, \frac 1n\}$$ thus $$n\ge q\implies f_n(\frac 1q)=\frac 1q$$ and $$\lim_{n\to+\infty}f_n(\frac 1q)=\frac 1q$$