Reminds me of the paradoxes of Zeno [1], especially the paradox of Achilles and the tortoise.

At least one can simply prove that 0.999... = 1 without much hard work. Maybe less controversial than the following:

    1 + 2 + 3 + ... [somehow] = -1/12 {{Riemann's zeta(-1)?}}
    1 + 2 + 4 + 8 + 16 + ... [somehow] = -1

As well as the weird prime product (Product of 1/(1-(p^-2)) for p prime) and the sum of x^-2 from x=1 to [sigh] being equal to (pi^2)/6 are some example of infinite beauty of mathematics that I remember.

[1]: https://en.wikipedia.org/wiki/Zeno's_paradoxes