On page 3 of http://www.math.dartmouth.edu/~carlp/Lehmer0.5.pdf the author write that the following inequalities follow from "the Hardy-Ramanujan inequality", but he doesn't point to a proof. The inequalities state that $$ \# \left \{ n \le t \mid \omega(n) \ge \lambda \log \log t \right \} = O \left ( \frac{e^{\lambda}t}{(\log t)^{1+\lambda \log (\lambda/e)}}\right ) $$ $$ \# \left \{ n \le t \mid \omega(n) \le \lambda \log \log t \right \} = O \left ( \frac{1}{(\log t)^{1+\lambda \log (\lambda/e)}}\right ) $$ hold uniformly for $\lambda \ge 1$ and $0 < \lambda \le 1$ respectively, where $\omega(n)$ is the function that counts the number of different prime divisors in $n$.
Can anyone help me to find a proof?