Question 99
a) Prove that $\frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots + \frac{1}{2^m} \lt m$, for all $m \in \mathbb{N}^*$.
b) Let $p_1, p_2, \dots, p_n$ be the sequence of the primes less than $2^{100}$. Prove that
$$
\frac{1}{p_1} + \frac{1}{p_2} + \dots + \frac{1}{p_n} \lt 10.
$$
b) Let $p_1, p_2, \dots, p_n$ be the sequence of the primes less than $2^{100}$. Prove that
$$
\frac{1}{p_1} + \frac{1}{p_2} + \dots + \frac{1}{p_n} \lt 10.
$$