The idea is that $\frac{a_{n+1}}{n!}$ tends to $a_1-1+e$, while $\frac{b_{n+1}}{n!}$ tends to $b_1+1-e$ (or something like that ). For sufficiently large $n$, the only common terms must be of the form $a_n=b_n$, because otherwise, if, for example, $a_n=b_m$ with $n<m$, this would mean that for some small $\varepsilon,\varepsilon'$ we would have $n!(a_1-1+e-\varepsilon)=m!(b_1+1-e-\varepsilon')$, and this is impossible for large $n<m$. On the other hand, $a_n=b_n$ for infinitely many $n$ leads to $a_1-1+e=b_1+1-e$, which is, again, impossible because $e$ is irrational.