Cantor gave several proofs of uncountability of reals; one involves the fact that every bounded sequence has a convergent subsequence (thus being related to the nested interval property). All his proofs are discussed here:

[MR2732322](https://mathscinet.ams.org/mathscinet-getitem?mr=2732322) (2011k:01009) 
Franks, John:
Cantor's other proofs that R is uncountable. (English summary)
Math. Mag. 83 (2010), no. 4, 283–289.