real analysis - Is the sequence of functions $f_n=chi_{[n,n+1]}$ uniformly integrable?

I wish to prove or disprove that the sequence of functions $f_n=\chi_{[n,n+1]}$ is uniformly integrable?

At a glance my judgement is YES, it is uniformly integrable.

From the definition of Uniform integrability, that's

A sequence ${f_n}$ is called uniformly integrable if $\forall \epsilon >0 \exists \delta > 0$ such that if $E \subset X$, $E$ measurable and $\mu (E)< \delta$ then $\forall n$ $\int_E |f_n| d\mu < \epsilon$.

So I let $E \subset R$ such that $\mu (E)<\delta$
then $\int_E|f_n|=\int|f_n|\chi_E \leq \mu (E)<\delta$.

So in this case $\epsilon =\delta$.

Does this make sense?