KindiakMath

Dini’s Mutated Result

Problem 1. Let \{f_n\} be a sequence of functions on [0, 1] that satisfy the following properties:

  • each f_n is non-decreasing on [0, 1],
  • f_n \to f pointwise,
  • f is continuous on [0, 1].

Prove that f_n \to f uniformly. This is a modified convergence criterion from the more famous Dini’s theorem.

(Click for Solution)

Solution. Fix \epsilon > 0. Using the (uniform) continuity of f, for any k_1 > 0, there exists \delta > 0 such that

|u-v| < \delta \quad \Rightarrow \quad |f(u) - f(v)| < k_1 \cdot \epsilon.

By the Archimedean property, find N_1 \in \mathbb N so that 1/N_1 < \delta. Partition [0, 1] into \{x_0, x_1,\dots,x_{N_1}\}, x_i = i/N_1. For each i, point-wise convergence f_n \to f yields that for any k_2 > 0, there exists N_2 \in \mathbb N such that whenever n > N_2,

n > N_2 \quad \Rightarrow \quad |f_n(x_i) - f(x_i)| < k_2 \cdot \epsilon,\quad i=0,1,2,\dots,N_1.

Finally, fix x \in [0, 1] and suppose x \in [x_0, x_1] without loss of generality. Then for n > N_2 + 1, point-wise monotonicity yields

\begin{aligned} |f_n(x) - f(x)| &\leq |f_n(x) - f_n(x_1)| + |f_n(x_1) - f(x_1)| + |f(x_1) - f(x)| \\ &< |f_n(x) - f_n(x_1)| + k_2 \cdot \epsilon + k_1 \cdot \epsilon \\&< |f_n(x) - f_n(x_1)| + (k_1 + k_2) \cdot \epsilon .\end{aligned}

To bound the first term, we use monotonicity to obtain

\begin{aligned} |f_n(x) - f_n(x_1)| &\leq |f_n(x_0) - f_n(x_1)|\\ &\leq |f_n(x_0) - f(x_0)| + |f(x_0) - f(x_1)| + |f(x_1) - f_n(x_1)| \\ &< k_2 \cdot \epsilon + k_1 \cdot \epsilon + k_2 \cdot \epsilon \\ &= (k_1 + 2 k_2 )\cdot \epsilon. \end{aligned}

Combining the estimates yields

\begin{aligned} |f_n(x) - f(x)| &< |f_n(x) - f_n(x_1)| + (k_1 + k_2) \cdot \epsilon \\ &< (2k_1 + 3k_2) \cdot \epsilon.\end{aligned}

Finally, set k_1 = 1/4, k_2 = 1/6 to obtain the desired result.

—Joel Kindiak, 31 Jan 25, 1422H

,

Published by

joelkindiak

Leave a comment