Let's say there are books lined up as shown in the figure.
We know if we push over one book, the rest of the books fall over as seen in the figure below.
A. If we push over one book, it should fall.
B. If a book is falling and has been placed correctly, it will knock over its neighbour. Though intuitively we can say that if the next to the last book falls, the last book also falls, but this needs to be proved by logical reasoning.Now let us consider the following.
1. Assume that there is some book 'k' which doesn't fall over i.e., k is the first book which behaves in a different manner.2. Since k is the first book, the book right before k must have fallen over.
3. But we know from B, a falling book always knocks over the next one.4. So book k will fall over and we have a contradiction.
We think of each book as an instance of a proposition. If a given instance is true, the corresponding book will fall over, given a sequence of instances (row of books).If we can prove
1. The proposition is true in the first instance.2. And if a given instance is true.
3. The next one in the sequence will also be true.Then the proposition will be true in all instances.
This is called proving a proposition by Induction.