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 (2011k:01009) Franks, John: Cantor's other proofs that R is uncountable. (English summary) Math. Mag. 83 (2010), no. 4, 283–289.

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 (2011k:01009) Franks, John: Cantor's other proofs that R is uncountable. (English summary) Math. Mag. 83 (2010), no. 4, 283–289.