 Subsequence selection as regular matrix summability

Selecting a subsequence is a regular matrix summability method: every row has one entry equal to one, every fixed column eventually vanishes, and all row sums equal one. Every bounded sequence of real or complex numbers has a convergent subsequence by the Bolzano-Weierstrass theorem, so each bounded sequence is summable by some sequence-dependent regular method. The order of the quantifiers matters: no single regular matrix sums all bounded sequences.