TITLE: What is meant by a "diagonalization argument"?
QUESTION [9 upvotes]: From Wikipedia:

A variety of diagonal arguments are used in mathematics. 

Cantor's diagonal argument
Cantor's theorem
Halting problem
Diagonal lemma


Besides the above four examples, there is another one I found in a blog. When proving that "if a sequence of measurable mappings converges in measure, then there is a subsequence converging a.e.", the construction of the subsequence is also called diagonalization:

The method for establishing this result is fairly typical of such
  arguments: we rely on diagonalization along with the control that
  Borel-Cantelli gives us. Let $f_n$ be a sequence that converges in
  measure to $f$. This means that for any n we have a $f_{m_n}$ with
  $\mu(|f_{m_n} - f| > 1/n) < 2^{-n} $. Applying Borel-Cantelli to the
  sequence  of sets $A_n = \{x | |f_{m_n}(x) - f(x)| > 1/n\}$ yields
  $\mu(\limsup_m \cup_{n=m}^\infty A_n) = 0$. But this is simply saying
  that the set of points on which $f_{m_n}$ doesn’t converge to $f$ has
  measure $0$.

As someone who has not been very much exposed to "diagonalization arguments", I wonder if the examples above have something in common, so that we may answer what "diagonalization argument" is and what kinds of problems it may help to solve?

REPLY [2 votes]: The Arzelà Ascoli theorem is an instance of this. See http://en.wikipedia.org/wiki/Arzel%C3%A0%E2%80%93Ascoli_theorem. What is interesting about this example is that this theorem has applications in differential equations. One example of an application is the theorem that in a hyperbolic surface, every free homotopy class of loops has a unique geodesic in that class.