Decimal expansion of an irrational number not ending in a particular sequence Consider the set $N$ of natural number, $N$={1,2,3,4,5,6,7,8,9,10,11...} and consider the subsequence  {$N_i$} ,$i \in N$ 

Each $N_i$ consists of elements in ascending sequence greater than $i$

$N_1$={2,3,4,.....}  ,$\quad$$N_2$={3,4,5,.....},$\quad$$N_{11}$={12,13,14,15,...} and so on

0.1234567......., $\quad$ 0.0123456.....,$\quad$ 0.1119012345679810....$\quad$ 0.992018151617181920 $\quad$ are are all irrational numbers in $[0,1] $, but the tail  of each sequence  can be mapped to a subset of {$N_i$}: for example $\quad$ 0.992018151617181920... can be mapped to $\quad$ $N_{15}$={15,16,17,18,19,20,21,....}

My question is can someone give example of irrational number in $[0,1] $ ,such that the tail of its decimal expansion cannot be mapped to any subset of {$N_i$} ?

Of course there are uncountable of such examples. But to construct one is not trivial. You may know the decimal expansion of $\pi$ upto one billion decimal places but that doesn't stops its remaining digits being mapped to {$N_i$}.
How about: 0.1121231234123451234561234567... ?
How about: 0.1121231234123451234561234567... ?