High Quality Content by WIKIPEDIA articles! In the mathematical field of analysis, Dini's theorem states that if X is a compact topological space, and { fn } is a monotonically increasing sequence (meaning fn( x ) ? fn+1( x ) for all n and x) of continuous real-valued functions on X which converges pointwise to a continuous function f, then the convergence is uniform. An analogous statement holds if { fn } is monotonically decreasing. 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. Note also that the limit function must be continuous, since a uniform limit of continuous functions is necessarily continuous. (if the limit function is not required to be continuous, the theorem doesn't hold, as seen by the sequence fn(x)=xn over [0,1]).

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