Question 1030129
Suppose C is infinite.

Then by the Bolzano-Weierstrass theorem, since C is bounded (and closed!), C will have a limit point {{{x[0]}}} that is contained in C.

This implies that there will be a positive integer n>1 such that {{{abs(abs(x-x[0])) < 1/n < 1}}} for an infinite number of elements x in C.

Contradiction, because it should be that {{{abs(abs(x-x[0])) >= 1}}}.

Hence C should be finite.