Heine–Cantor theorem
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In mathematics, the Heine–Cantor theorem, named after Eduard Heine and Georg Cantor, states that if M is a compact metric space, then every continuous function
- f : M → N,
where N is a metric space, is uniformly continuous.
For instance, if f : [a,b] → R is a continuous function, then it is uniformly continuous.
This is not Cantor's theorem.
[edit] Proof
Suppose that f is continuous on a compact metric space M but not uniformly continuous, then the negation of
such that 
for all x, y in M
is:
such that 
such that 
and 
.
where d and ρ are the distance functions on metric spaces M and N, respectively.
Choose two sequences xn and yn such that
as the metric space is compact there exists two converging subsequences (
to x0 and 
to y0), so
but as f is continuous and and converge to the same point, this statement is impossible.
