Convergence proof techniques are canonical patterns of mathematical proofs that sequences or functions converge to a finite limit when the argument tends to infinity.
There are many types of sequences and modes of convergence, and different proof techniques may be more appropriate than others for proving each type of convergence of each type of sequence. Below are some of the more common and typical examples. This article is intended as an introduction aimed to help practitioners explore appropriate techniques. The links below give details of necessary conditions and generalizations to more abstract settings. Proof techniques for the convergence of series, a particular type of sequences corresponding to sums of many terms, are covered in the article on convergence tests.