One set of 20 plus six sets of 3 add up to 38, so that proves this proposition is true in the first case.
In mathematical induction one shows how to get from any instance to the next, e.g. from 38 to 39, then from 39 to 40, then from 40 to 41, and so on. But one must show it not just concretely in several instances, nor are a trillion instances enough. Rather one must show it works in all infinitely many cases.
Suppose an order for n bottles can be filled with either:
- at least two sets of 20 and possibly some sets of 3, or
- at least one set of 20 and at least six sets of 3, or
- at least 13 sets of 3.
Then an order for n+1 bottles rather than only n bottles can be filled with either:
- at least two sets of 20 and possibly some sets of 3, or
- at least one set of 20 and at least six sets of 3, or
- at least 13 sets of 3.
In the first case (at least two sets of 20 and possibly some sets of 3) one can delete a set of 20 from the shipment and add seven sets of 3. Thus one has deleted 20 bottles and added 21, so the total number of bottles shipped is now n+1 rather than n. And now one has at least one set of 20 and at least six sets of 3 (since one has in fact at least seven sets of 3).
In the second case, one can likewise delete a set of 20 and add seven sets of 3, and again we have increased the size of the shipment by 1, and now we have at least 13 sets of 3.
In the third case, one can delete 13 sets of 3 from the order and add two sets of 20. This decreases the number of bottles by 39 and then increases it by 40, so again it has 1 more than before. And this puts us in the situation where there are at least two sets of 20.
Thus we can always increase the size of the shipment by 1, by doing one of the three things stated above.
So we can increase it from 38 to 39, then from 39 to 40, then from 40 to 41, and so on.
