Probe the following sequences is null

Let $\epsilon>0$. We want to file $N_0$ so that for all $n\geq N_0$ we have $|A_n|<\epsilon$, i.e. 
\[|\frac{(-1)^{n+1}}{8n^3-3}|= \frac{1}{8n^3-3}<\epsilon\]
\[\Rightarrow     8n^3-3>\frac{1}{\epsilon}  \Rightarrow n^3> \frac{1}{8}(\frac{1}{\epsilon}+3)\]
\[\Rightarrow n> \sqrt[3]{\frac{1}{8}(\frac{1}{\epsilon}+3)}. \]
So we can choose $N_0$ to be an integer such that 
\[N_0> \sqrt[3]{\frac{1}{8}(\frac{1}{\epsilon}+3)},\]
then for $n>N_0$ we have
\[|A_n|<\epsilon.\]
This means that $A_n$ is a null sequence.