Let's provide an example of a proof by contradiction for the somehow dry explanations above:

You might want to prove that there is an infinite amount of rational numbers (i.e. numbers of the form x/y with x and y being positive integers) between 0 and 1 (inclusive) {This is the hypothesis mentioned above}. For a proof by contradiction we assume that there was only a finite amount of rational numbers between 0 and 1 {negation of the hypothesis}. But if there only was a finite amount, we must be able to order the numbers. If we then counted the ordered rational numbers starting from 1, we must sometime reach 0 independent of the counting scheme we use {We are trying to deduce something}. So as a counting scheme we choose to count all the rational numbers 1/x (with x being a positive integer), which nicely counts an ordered set of rational numbers between 1 and 0.

Alas, for no x 1/x will reach 0 (no, "infinite" is not an integer :). Therefore, there is at least one counting scheme which will never reach 0 {the contradiction...}, which violates the implications we deduced from having only a finite amount of rational numbers between 0 and 1. But if the implications are wrong, so has to be the assumption (see implication and contrapositive). Therefore there is an infinite amount of rational numbers between 0 and 1 {... which establishes the truth of the hypothesis}. qed

BTW, if you also take real numbers into account, there is not only an infinite amount but an uncountable amount of numbers between 0 and 1 (see Real Numbers are Uncountable - proof).

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