\begin{align*}
||M_m|| &= \frac{1}{N}\max_{k=1, \ldots, N} \left(1 - \left(1-\frac{\widehat{\mu}_k}{\widehat{\mu}_k + \lambda}\right)^m\right)^2 \widehat{\mu}_k^{-1} \\  &\stackrel{(i)}{\leq}
\frac{1}{N}\max_{k=1, \ldots, N} \min\left\{1, m^2 \left(\frac{\widehat{\mu_k}}{\widehat{\mu_k} + \lambda}\right)^2\right\} \widehat{\mu_k}^{-1}  \\
&\leq \frac{1}{\lambda^2  N } \max_{k=1, \ldots, N} \min\left\{\widehat{\mu_k}^{-1}, m^2 \widehat{\mu_k}\right\} \leq \frac{m^2 \widehat{\mu}_1}{\lambda^2 N } \rightarrow 0,
\end{align*}
