First of all, we remove
$100$ digits whatsoever and we cannot change any digit place, so in order to get the biggest or smallest number we need to play with the first digits as big/small as possible. Since 9 is the biggest digit, to make it biggest, we need to try to get as many 9-digit as possible, if somehow it is not possible to get 9 by removing the digits (it will happen examplified below), we need to consider the next biggest digit 8 and etc....
To get 9, we need to remove first 8 digits from 123456789101112...,
Remove every 19 digits after 9 because the next 9 is after 19 digits, then look for another 9 and continue removing...
and our number becomes something like below after removing 84 digits:
and we have
16 digit left to remove but we cannot reach to 9 because the next 9 is 19 digits after like before... so we should consider getting 8 in 16 digit, can we reach to 8 with 16 digits? no, then 7? yes after 15 digits luckily..!
remove 15 digits again
then our number becomes:
99999975859..... with 1 digit removing option!
remove $5$ which is between $7$ and $8$, since we dont have 9 after 1 digit, only 8 is biggest possible number!
then the number becomes
For the smallest one, the same logic is applicable,
Remove numbers until we encounter $0$.
The frequency of
$0$ in the sequence is 19 again
so our number becomes
Then we have 15 digits left to remove so with the same principle
if we remove $15$ digits, we will not able to reach $0$, then we should look for $1$.
$1$ exists in the next digit, so remove 1 digit only, then look for another one for the 14 digits if we cant find $1$, look for $2$ etc... this is the general methodology to find the biggest or smallest number.
So our number becomes (if I did not mess up)