Justify that the function $\psi_n$ is differentiable on $\mathbb{R}_+$ and that $\psi_n' = m_n$. We admit that, when $\psi$ is differentiable on $\mathbb{R}_+^*$, then $(\lim \psi_n)' = \lim \psi_n'$, that is $\psi' = m$, on $\mathbb{R}_+^*$.
Justify that the function $\psi_n$ is differentiable on $\mathbb{R}_+$ and that $\psi_n' = m_n$.
We admit that, when $\psi$ is differentiable on $\mathbb{R}_+^*$, then $(\lim \psi_n)' = \lim \psi_n'$, that is $\psi' = m$, on $\mathbb{R}_+^*$.