a question on prime numbers and infinite series

prove that the infinite sum - $∑(1/2)^p$, where $p$ runs over all the $prime$ numbers,is $irrational$

one idea that may work is to use the given lemma

$Lemma:$ $α$ is an irrational number iff there exists two convergent integer sequences ${a_n}$ and $b_n$ such that $(a_n-αb_n)≠0$ for all $n$. but $lim(a_n-αb_n)=0$

The proof is just by contradiction by assuming $α$ to be rational.I have tried to work out this lemma but failed .Plz try this.

Answer

Hint: look at the similar number $$\sum_{i = 1}^\infty \left(\frac{1}{10}\right)^{p_i} \approx 0.0110101000101000101$$ (be sure to notice that's "approximately," not "equals").

Clearly the $1$s will occur in the prime positions. You should already know that there are infinitely many primes. You should also know that their distribution feels kind of random. With one exception, any two $1$s are separated by an odd number of $0$s. But other than that... kind of random.

If this number was rational, we could find integers $m$ and $n$ such that one divided by the other gives this number. Try truncating the number at the prime positions. That does give rational numbers, but... $$\frac{1}{100}, \frac{11}{1000}, \frac{1101}{100000}, \frac{110101}{10000000}, \ldots$$