An easy example of a non-constructive proof without an obvious “fix”?

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).

Answer

Some digit occurs infinitely often in the decimal expansion of $\pi$.

Attribution
Source : Link , Question Author : Valentin Golev , Answer Author : Neil Strickland

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