Let $A_n=\{\frac{1}{n}\}$ for all $n\in\mathbb{N}$.
I have to show that \begin{equation} \bigcup_{n\in\mathbb{N}}A_n=\{1,\frac{1}{2},\frac{1}{3}\dots\} \end{equation} is Peano-Jordan measurable.
Indicate with $m_i$ the inner Peano-Jordan measure and with $m_e$ the outer Peano-Jordan measure. Now, since \begin{equation} \bigcup_nA_n\subset[0,1] \end{equation}
$m_e\big(\bigcup_nA_n\big)\le m_e([0,1])=1$ $\quad$ and$\quad$$m_i\big(\bigcup_nA_n\big)\le m_i([0,1])=1$.
From the definition \begin{equation} m_i\big(\cup_n A_n\big)\le m_e\big(\cup_n A_n\big). \end{equation}
From these considerations, how can I show that $\cup_n A_n$ is Peano-Jordan measurable?
Thanks!