Why is $f_n(x) = x^n$ not uniformly convergent on $(0, 1)$?

Hint. You have $$\sup_{x\in(0,1)}f_n(x)=1.$$


Note that $\lim_{x\to1} f_n(x) = 1$ for all $n$. This breaks uniform convergence because we can get close enough to $1$ such that $f_N(x) > \frac12$ for any fixed $N$.


Hint If $x_n =1-\frac{1}{n}$ then $f_n(x_n) \to \frac{1}{e}$.

Use this to contradict $d(f_n(x_n), f(x_n))<\epsilon$.

Tags:

Calculus

Sequences And Series

Convergence Divergence

Uniform Convergence

Related

In which algebraic setting can I state (and prove) the binomial theorem? I want to learn mathematics to extend myself. Is the axiom of choice really all that important? Is power set of a power set of a set equal to the power set of the same set? Technique for proving four given points to be concyclic? Given $\frac{\log x}{b-c}=\frac{\log y}{c-a}=\frac{\log z}{a-b}$ show that $x^{b+c-a}\cdot y^{c+a-b}\cdot z^{a+b-c} = 1$ Proving that $\frac{1}{n}+\frac{1}{n+1}+\cdots+\frac{1}{2n}>\frac{13}{24}$ by induction. Where am I going wrong? The image of a path-connected set under a continuous map is path-connected Can someone explain 4th dimensional objects? Finding triplets $(a,b,c)$ such that $\sqrt{abc}\in\mathbb N$ divides $(a-1)(b-1)(c-1)$ What is a good reference for rigorous Electromagnetism and Electrodynamics? What is this pattern called?

Recent Posts