I wanted to give an easy example of a non-constructive proof, or, more precisely, of a proof which states that an object exists, but gives no obvious recipe to create/find it.
Euclid's proof of the infinitude of primes came to mind, however there is an obvious way to "fix" it: just try all the numbers between the biggest prime and the constructed number, and you'll find a prime in a finite number of steps.
Are there good examples of simple non-constructive proofs which would require a substantial change to be made constructive? (Or better yet, can't be made constructive at all).
Best Answer
Some digit occurs infinitely often in the decimal expansion of $\pi$.