Find one irrational number between

Let 

$a=0.2101$

$b=0.2222 \ldots=0 . \overline{2}$

Here a and b are rational numbers .Since a has terminating and b has repeating decimal. We observe that in second decimal place a has 1 and b has 2. So a < b.

Hence one irrational number is $0.220100100010000 \ldots$ lying between $0.2101$ and $0.2222 \ldots$