Exercise about a sequence of functions in $L^2(0, +\infty)$

Question

In $L^2(0, +\infty)$, consider the sequence of functions $$ f_n(x)=\begin{cases}1, \ x\in[n, n+1]\\0,\ x\notin[n, n+1]\end{cases}. $$ with $n\in\mathbb{N_0}$. Say if the sequence is a Cauchy sequence or not and prove that the set $\{f_n\}_{n\in\mathbb{N}_0}$ is orthonormal but not complete in $L^2(0, +\infty)$

My attempt. To say if the sequence is Cauchy or not, I compute $$ \|f_n-f_m\|^2=\int_{0}^{\infty}|f_n(x)-f_m(x)|^2dx=0,\ \ \ \ n\neq m $$ but the solution of the exercise gives $2$ as result of this integral. Is it correct? What about the completeness of the set?

Thank You

Answer 1

If $n\neq m$,

$$\int_0^\infty|f_n(x)-f_m(x)|^2dx=\int_n^{n+1}|1|^2dx+\int_m^{m+1}|-1|^2dx=1+1=2$$