In the mathematical field of analysis, Dini's theorem says that if a monotone sequence of continuous functions converges pointwise on a compact space and if the limit function is also continuous, then the convergence is uniform.
is a compact topological space, and
is a monotonically increasing sequence (meaning
) of continuous real-valued functions on
which converges pointwise to a continuous function
, then the convergence is uniform.
The same conclusion holds if
is monotonically decreasing instead of increasing.
The theorem is named after Ulisse Dini.
[2] This is one of the few situations in mathematics where pointwise convergence implies uniform convergence; the key is the greater control implied by the monotonicity.
The limit function must be continuous, since a uniform limit of continuous functions is necessarily continuous.
The continuity of the limit function cannot be inferred from the other hypothesis (consider
ε > 0
be the set of those
( x ) < ε
is open (because each
is the preimage of the open set
( − ∞ , ε )
, a continuous function).
is monotonically increasing,
is monotonically decreasing, it follows that the sequence
is ascending (i.e.
converges pointwise to
, it follows that the collection
is an open cover of
By compactness, there is a finite subcover, and since
are ascending the largest of these is a cover too.
Thus we obtain that there is some positive integer
is a point in
< ε