aman88 wrote:Thanks but isn't it important to consider how many integers are there in set K? Say, if we have 10 integers in set K, then there will not be a 100 in it. And if we have 50 or more, then there will be 100. So ambiguous! How did we reach to a conclusion that there will be infinite integers?

Thanks.

Set K must contain an infinite number of values. We know this because there cannot be a greatest integer in set K.

For a moment, let's pretend that there is a greatest integer in set K, and then we'll demonstrate that this scenario would be impossible.

Aside: this method of reasoning is called "Proof by contradiction" (see: https://en.wikipedia.org/wiki/Proof_by_contradiction for more info)

Okay, let's say that N is the greatest integer in set K.
The question tells us that if integer r is in the K, then r+1 is also in K.
So, if N is in set K, then N+1 must be in set K.
If N and N+1 are both in set K, then N cannot be the greatest integer in set K.
But, we began by defining N as the greatest integer in set K.
This contradiction tells us that there cannot be a greatest integer in set K, which means set K contain an infinite number of integers.

Cheers,
Brent