Given that the probability distribution is:
\begin{equation*} X \sim \begin{pmatrix} x_1 & x_2 & x_3 & \dots & x_N \\ p_1 & p_2 & p_3 & \dots & p_N \\ \end{pmatrix} \end{equation*}
with $p_1 \leq p_2 \leq \dots \leq p_N$. Prove that:
$$ -\sum_{i=1}^{N}p_i\log_2 p_i \geq 2(1-p_N) $$
The hint in the exercice is to use $$ -\sum_{i=1}^{N}p_i\ln p_i \geq (1-p_N) $$
for $p_N \geq 0.5$, and
$$-\sum_{i=1}^{N}p_i\log_2 p_i \geq -\log_2p_N $$
for $p_N \leq 0.5$. But I didn't understood very well this hint.