This question is a bit confusing. I decided to start by showing that all cards don't contain the same number.
Lets say I pick a subset of $10$ cards then they have $10-19$
$10+11+12+13+..+18+19=145$ which is not divisible by $100$
but to be divisible by $100$, each card has to end in $0$ that leads to each card being $10,20,30,...80,90$ but that doesn't guarantee divisibility with the subsets.
If each card is $10$ and i take a subset containing $14$ cards, then it still wont be divisible by $100$
Any ideas on this problem?
The assumptions are impossible to satisfy: The sum of the empty subset of the cards is $0$ which is divisible by $100$.
The claim is therefore vacuously true. Q.E.D.
(It would have been more involved if the assumption said "... any nonempty subset of cards ...")