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?
 A: The approach is correct with this definition of uniform integrability. 
A concern about this one is that a uniform integrable family with this definition may not contain an integrable function. For example, the family consisting of the constant function equal to $1$ would be uniformly integrable. 
A common definition is that a family $\mathcal F$ of function on a measure space $\left(X,\mathcal A,\mu\right)$ is uniformly integrable if for each positive $\varepsilon$, there exists an integrable function $g$ such that
$$
\sup_{f\in\mathcal F}\int_{\left\{\left\lvert f\right\rvert\gt g\right\}}\left\lvert f\right\rvert\mathrm d\lambda\lt \varepsilon.
$$ 
With this definition, the family $(f_n)$ defined by $f_n=\chi_{[n,n+1]}$ is not uniformly integrable. Indeed, let $\varepsilon=1/2$. If a function $g$ is such that $$\sup_{n\geqslant 1}\int_{\left\{\left\lvert f_n\right\rvert\gt g\right\}} f_n\mathrm d\lambda\lt 1/2$$
then $\lambda\left(\{g\lt 1\}\cap [n,n+1]\right)\lt 1/2$ hence $\lambda\left(\{g\geqslant 1\}\cap [n,n+1]\right)\geqslant 1/2$ for all $n$ hence $g$ cannot be integrable.
