this is was a homework exercise for one of my classes, and there was a hint to use a diagonal argument.

i’ve not seen a diagonal argument before aside from Cantor’s uncountability of the reals from real analysis. thus my strategy is to try and mimic it, as follows:

seeking a contradiction, i assume there is a countable basis x1, x_2, x_3, … where x_i = (a{i,1}, a{i,2}, a{i,3}, … )

now, following Cantor’s argument for the reals, i construct a signal x = (a{1,1} 1, a{2,2} 1, a_{3,3} 1, … )

here’s where i am stuck: how do i show that x cannot be expressed as a linear combination of basis signals, giving me my desired contradiction?