Irrationality of an "Euler-like" number

Question

Let $(a_n)_{n=0}^{\infty}$ be a sequence of zeroes and ones such that $a_n=1$ for infinitely many $n$. Let $\displaystyle x:=\sum_{n=0}^{\infty} \frac{a_n}{n!} .$

Is $x$ irrational? I believe it is, but I don't know how to prove it. I'll appreciate any help.

Answer 1

As was pointed out by leoli1, this problem can be easily solved by slightly modifying Fourier's proof of irrationality of Euler's number.

Case closed.