= Subsequence selection as regular matrix summability
{title2=$a_{ij}=\mathbf1_{\{j=n_i\}},\quad n_i\uparrow\infty$}
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.
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