Cousin's theorem

In real analysis, a branch of mathematics, Cousin's theorem states that: This result was originally proved by Pierre Cousin, a student of Henri Poincaré, in 1895, and it extends the original Heine–Borel theorem on compactness for arbitrary covers of compact subsets of

However, Pierre Cousin did not receive any credit.

Cousin's theorem was generally attributed to Henri Lebesgue as the Borel–Lebesgue theorem.

Lebesgue was aware of this result in 1898, and proved it in his 1903 dissertation.

[1] In modern terms, it is stated as: Cousin's lemma is studied in reverse mathematics where it is one of the first third-order theorems that is hard to prove in terms of the comprehension axioms needed.

Cousin's theorem is instrumental in the study of Henstock–Kurzweil integration, and in this context, it is known as Cousin's lemma or the fineness theorem.

A gauge on

is a strictly positive real-valued function

, while a tagged partition of

is a finite sequence[2][3] Given a gauge

and a tagged partition

denotes the open ball of radius

Cousin's lemma is now stated as: Cousin's theorem has an intuitionistic proof using the open induction principle, which reads as follows: An open subset

of a closed real interval

is said to be inductive if it satisfies that

The open induction principle states that any inductive open subset

must be the entire set.

be the set of points

-fine tagged partition on

is open, since it is downwards closed and any point in it is included in the open ray

to handle the base case) we have a partition of length

{\displaystyle x_{n}>\mathrm {max} (a,r-{\tfrac {1}{2}}\delta (r))}

, we may form a partition of length

To show this, we split into the cases

In the first case, we set

In both cases, we can set

and obtain a valid partition.

By open induction,

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