Dini's theorem

Dini's theorem 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.[1] Contents 1 Formal statement 2 Proof 3 Notes 4 References Formal statement If {displaystyle X} is a compact topological space, and {displaystyle (f_{n})_{nin mathbb {N} }} is a monotonically increasing sequence (meaning {displaystyle f_{n}(x)leq f_{n+1}(x)} for all {displaystyle nin mathbb {N} } and {displaystyle xin X} ) of continuous real-valued functions on {displaystyle X} which converges pointwise to a continuous function {displaystyle fcolon Xto mathbb {R} } , then the convergence is uniform. The same conclusion holds if {displaystyle (f_{n})_{nin mathbb {N} }} 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.

Proof Let {displaystyle varepsilon >0} be given. For each {displaystyle nin mathbb {N} } , let {displaystyle g_{n}=f-f_{n}} , and let {displaystyle E_{n}} be the set of those {displaystyle xin X} such that {displaystyle g_{n}(x)N} and {displaystyle x} is a point in {displaystyle X} , then {displaystyle |f(x)-f_{n}(x)|

Si quieres conocer otros artículos parecidos a Dini's theorem puedes visitar la categoría Theorems in real analysis.