1. Proof That There Is No Smallest Number
Claim:
There is no positive number X that is closer to zero in value than all other positive numbers. In other words, given any real X>0, there will always be a smaller real value Y such that 0 < Y < X.
We will prove this by a method called Proof By Contradiction
We assume the premise X>0 is true and the contradiction of the conclusion is true (there is no smaller value Y) and then show that a contradiction results.
For any positive X we pick, no matter how small, how could we always get a smaller value that is still positive?

2. Solution
Given X>0 is any number close to zero, letting Y = X/2 still results in a positive number and also 0 < Y < X. This contradicts our assumption that no such Y value exists and proves the original claim that there is no smallest positive real number.
This method of proof is often used when direct methods will not work. Albert Einstein started his studies at a young age doing such proofs in Euclidean Geometry. For more information on this, see
Einstein's Proof