Mathematically, Dini's theorem is expressed as follows: Hypothesis? x？ Is it a compact topological space? F(n) is? x？ A monotonically increasing sequence of continuous real functions (even if it is for any? n？ And then what? x？ Really? x？ Both). What if this function sequence converges point by point to a continuous function? f？ , then this function column converges uniformly to? f？ . This theorem is named after the Italian mathematician Julius Dini.

The relevant contents of Dini theorem are as follows:

Form of function sequence: Let (x, r) be a compact topological space, and {fn} be a continuous real function sequence on x and satisfy the following two conditions:

1) {fn} converges to a continuous function f point by point,

2) {fn(x} is a monotone sequence at every point,

Then there is an open neighborhood for any point in X to make it converge to this neighborhood uniformly. It is known that there are finite such open neighborhoods covering the whole X, namely:

Extended data:

This theorem is also applicable to monotone decreasing function sequences. Because of the stronger monotonicity condition, this theorem is one of the few examples where uniform convergence can be deduced from point-by-point convergence.

Note that the f in the theorem must be continuous, otherwise a counterexample can be constructed. For example, the function column {x} on the interval [0, 1]. This is a monotonically decreasing function, which converges to the function f point by point: when x belongs to [0, 1], f(x) equals 0, and f( 1) equals 1. But this function sequence is not uniformly convergent because f is discontinuous.

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