# Pointwise limit function $f$ of sequence $(f_n)$

My sequence of functions $$f_n (x) = \begin{cases} 1 & ,x = \frac{1}{n} \\ x & ,x = 1,1/2, ...,1/(n-1) \\ 0 & ,otherwise \end{cases}$$

My attempt is to fix $$k \in \mathbb{N}$$, consider the following cases when $$x = 1/k$$ for $$n \geq k$$ and $$x \neq 1/k$$. Is there a better approach to find the pointwise limit $$f$$ of this sequence $$f_n$$?

• Pointwise limits must be calculated pointswise. So, yes, you have to split up in these cases. – user370967 Nov 19 '18 at 17:59

Let $$x$$ be a real.
If $$x=\frac 1k$$ then for large enough $$n\ge k+2,$$ we will have
$$\frac 1n <\frac{1}{n-1} then $$f_n(x)=x$$
and if $$x\ne \frac 1k \implies f_n(x)=0$$
The pointwise limit function is $$f:x\mapsto x$$ if $$x=\frac 1k$$ and zero elsewhere.