Normal order

Template:Distinguish In mathematics, the Heine–Cantor theorem, named after Eduard Heine and Georg Cantor, states that if f : M → N is a continuous function between two metric spaces, and M is compact, then f is uniformly continuous. An important special case is that every continuous function from a closed interval to the real numbers is uniformly continuous.

Proof

Uniform continuity for a function f is stated as follows:

$\mathrm{\forall }\epsilon >0\mathrm{\exists }\delta >0\mathrm{\forall }x,y\in M:(d_M(x,y)<\delta \Rightarrow d_N(f(x),f(y))<\epsilon ),$

where d_M, d_N are the distance functions on metric spaces M and N, respectively. Now assume for a contradiction that f is continuous on the compact metric space M but not uniformly continuous; in this case, the negation of uniform continuity for f is that

$\mathrm{\exists }{\epsilon }_0>0\mathrm{\forall }\delta >0\mathrm{\exists }x,y\in M:(d_M(x,y)<\delta \wedge d_N(f(x),f(y))\ge {\epsilon }_0).$

Fixing ε₀, for every positive number δ we have a pair of points x and y in M with the above properties. Setting δ = 1/n for n = 1, 2, 3, ... gives two sequences {x_n}, {y_n} such that

$d_M(x_n,y_n)<\frac{1}{n}\wedge d_N(f(x_n),f(y_n))\ge {\epsilon }_0.$

As M is compact, the Bolzano–Weierstrass theorem shows the existence of two converging subsequences ($x_{n_k}$ to x₀ and $y_{n_k}$ to y₀) of these two sequences. It follows that

$d_M(x_{n_k},y_{n_k})<\frac{1}{n_k}\wedge d_N(f(x_{n_k}),f(y_{n_k}))\ge {\epsilon }_0.$

But as f is continuous and $x_{n_k}$ and $y_{n_k}$ converge to the same point, this statement is impossible. The contradiction proves that our assumption that f is not uniformly continuous cannot be true, so f must be uniformly continuous as the theorem states.

For an alternative proof in the case of M = [a, b] a closed interval, see the article on non-standard calculus.

External links

• Template:Planetmath reference
• Template:Planetmath reference

Template:Mathanalysis-stub