Definition
Cantor's diagonal argument is a proof technique introduced by Georg Cantor in 1891 to show that the real numbers are uncountable. Given any proposed list of reals, the argument constructs a real that differs from each listed real in at least one digit — by taking the n-th listed real's n-th digit and changing it. The constructed real cannot be on the list. Therefore no list contains all reals.
Why it matters
How it works
Suppose, for contradiction, the reals can be listed as r₁, r₂, r₃, .... Each rᵢ has a decimal expansion. Construct a new real d whose n-th digit differs from the n-th digit of rₙ (and is not 0 or 9, to avoid representation ambiguity). Then d differs from r₁ in its first digit, from r₂ in its second digit, and so on — d differs from every listed real and is not on the list. Contradiction; no such list exists. The reals are uncountable.