Construction of a sequence of measurable function

Question

Is there any sequence $f_n:\mathbb{R} \to \mathbb{R}$ of measurable functions such that for all $x \in \mathbb{R},\lim_nf_n(x)=x$ such that there exist a nonempty bounded interval $C_n=[\alpha_n,\beta_n]$ where $f_n$ is zero?

Answer 1

For instance:$$f_n(x)=x\left(1-\mathsf1_{[0,\frac1{n}]}(x)\right)$$Addendum:

Another example is:$$f_n(x)=x\left(1-\mathsf1_{[n,e^n]}(x)\right)$$ with shifting interval increasing in length.