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Proposition 1 Prove ${\sqrt{2}}$ is irrational.

Here is a proof using a traditional method (See Euclid’s Elements Book X which incorporates Theatetus work on incommensurable numbers. It includes a proof that ${\sqrt{2}}$ is irrational (Proposition 22), and ends with a proof that there are infinitely many distinct irrational numbers (Proposition 115): Read the rest of this entry »

This is a typical proof one may see in a undergraduate introduction to proofs mathematics course.

Proposition 1 Let the constant ${R}$ be defined such that

$\displaystyle R:=\frac{1}{F_{1}}+\frac{1}{F_{2}\cdot F_{1}}+\frac{1}{F_{3}\cdot F_{2}\cdot F_{1}}+\frac{1}{F_{4}\cdot F_{3}\cdot F_{2}\cdot F_{1}}\cdots$

and ${F_{n}}$ are the Fibonacci numbers where ${F_{n+k}>F_{k}}$ for ${n>0}$. Then ${R}$ is irrational.