[Math] Proof that there are infinitely many irrational numbers!

irrational-numbers

I want to prove that there are infinitely many irrational numbers! How can I do that?

I don't know where or how to start so any hint is appreciated.

Thanks! 🙂

Best Answer

HINT: Show that there is an injection from an infinite set, e.g. $\Bbb N$, into the set of irrational numbers. That is, send every natural number $n$ to some irrational number. Also, recall that if we take a non-zero rational number, its sum and product with any irrational number is irrational.

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