**Solution 6**

We can write where denotes the integral part of

Now, we can say that as

Let where is odd. Then

All such integers, must satisfy the property where is a non-negative integer.

Then

Now, Means, is even.

So

Or, is an integer.

Hence, is an integer for all positive reals satisfying

for some non-negative integer

As takes infinitely many values, number of such positive real numbers is also infinite.

This completes the proof.

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