Proof by contradiction and the well ordering principle

Question

I have a question regarding proof by contradiction and the well ordering principle on integers.

Okay lets says for example an arbitrary value $p$ is a positive integer and which I assume is the smallest positive integer for some property.

Then lets say I find an arbitrary value $r$, also a positive integer, which I calculate to be less than $p$ for the same property. Then I have a contradiction.

So my question is how is this possible. Lets substitute $1$ for $p$ and it satisfies the property, then how can I find $r$ since there is no smaller positive integer? I'm not sure if $0$ is considered a positive integer but if it is then substitute $p$ for $0$.

Answer 1

The point of the proof by contradiction is that you can find no such $r$. That's the contradiction. You assume something you want to be false, then prove something that can't be possible. It may be helpful to give a specific example of a proof that's confusing you.