Let $a_n(t)=\frac{1}{n} \text{1}_{ [0,n)}(t)$. Does this sequence converge in $L^1/$pointwise?
My ideas: We have for all $t$: $\mathbf{1}(t)\in\{0,1\}$ and $$0 \leq \frac{1}{n}\textbf{1}(t) \leq 1 \Rightarrow 0 \leq \left| \frac{1}{n}\textbf{1}(t)\right|^p \leq 1 \quad \forall p\ge1$$
So it's a null sequence and we have dominated convergence with dominated function $g(t)=1$. $g(t)=1$ is integrable so all $a_n$ is integrable for all $n$.
Point-wise it clearly converges to zero.
In the $L^1-$norm, it DOES NOT converge, since $$ \|a_{2n}-a_n\|_{L^1}=1 $$
and hence it is not a Cauchy sequence.