A finite sum of prime reciprocals How can you prove that $\sum\limits_k \frac1{p_k}$, where $p_k$ is the $k$-th prime, does not result in an integer?

Inspiring by this: https://en.wikipedia.org/wiki/Divergence_of_the_sum_of_the_reciprocals_of_the_primes#Partial_sums
$\displaystyle \frac{1}{p_1} + \frac{1}{p_2} + \cdots + \frac{1}{p_n} = \frac{1}{2} + \frac{1}{odd} + \cdots + \frac{1}{odd} = \frac{odd.odd \dots odd + 2.odd \dots odd + \cdots + 2.odd \dots odd}{2.odd \dots odd}$
$\displaystyle = \frac{odd + even + \dots + even}{2.odd} = \frac{odd + even}{even} = \frac{odd}{even} \neq integer$.
