In mathematics, there are many senses in which a sequence or a series is said to be convergent. This article describes various modes (senses or species) of convergence in the settings where they are defined. For a list of modes of convergence, see Modes of convergence (annotated index)

Note that each of the following objects is a special case of the types preceding it: sets, topological spaces, uniform spaces, TAGs (topological abelian groups), normed spaces, Euclidean spaces, and the real/complex numbers. Also, note that any metric space is a uniform space.

Convergence can be defined in terms of sequences in first-countable spaces. Nets are a generalization of sequences that is useful in spaces which are not first countable. Filters further generalize the concept of convergence.

In metric spaces, one can define Cauchy sequences. Cauchy nets and filters are generalizations to uniform spaces. Even more generally, Cauchy spaces are spaces in which Cauchy filters may be defined. Convergence implies "Cauchy-convergence", and Cauchy-convergence, together with the existence of a convergent subsequence implies convergence. The concept of completeness of metric spaces, and its generalizations is defined in terms of Cauchy sequences.

Erich Alfred Hartmann (19 April 1922 – 20 September 1993), nicknamed "Bubi" (the hypocoristic form of "young boy") by his comrades and "The Black Devil" by his Soviet enemies, was a German World War II fighter pilot and is the highest-scoring fighter ace in the history of aerial warfare. He claimed 352 aerial victories (of which 345 were won against the Soviet Air Force, and 260 of which were fighters) in 1,404 combat missions. He engaged in aerial combat 825 times while serving with the Luftwaffe. During the course of his career, Hartmann was forced to crash-land his damaged fighter 14 times. This was due to damage received from parts of enemy aircraft he had just shot down or mechanical failure. Hartmann was never shot down or forced to land due to fire from enemy aircraft.

Hartmann, a pre-war glider pilot, joined the Luftwaffe in 1940 and completed his fighter pilot training in 1942. He was posted to Jagdgeschwader 52 (JG 52) on the Eastern front and was fortunate to be placed under the supervision of some of the Luftwaffe's most experienced fighter pilots. Under their guidance, Hartmann steadily developed his tactics, which earned him the coveted Ritterkreuz mit Eichenlaub, Schwertern und Brillanten (Knight's Cross of the Iron Cross with Oak Leaves, Swords and Diamonds) on 25 August 1944 for claiming 301 aerial victories.