Cauchy's limit theorem

Cauchy's limit theorem, named after the French mathematician Augustin-Louis Cauchy, describes a property of converging sequences. It states that for a converging sequence the sequence of the arithmetic means of its first ${\displaystyle n}$ members converges against the same limit as the original sequence, that is ${\displaystyle (a_{n})}$ with ${\displaystyle a_{n}\to a}$ implies ${\displaystyle (a_{1}+\cdots +a_{n})/n\ \to a}$. The theorem was found by Cauchy in 1821, subsequently a number of related and generalized results were published, in particular by Otto Stolz (1885) and Ernesto Cesàro (1888).