- #1
nateHI
- 146
- 4
Homework Statement
Let ##O## be a proper open subset of ##\mathbb{R}^d## (i.e.## O## is open, nonempty, and is not equal to ##\mathbb{R}^d##). For each ##n\in \mathbb{N}## let
##O_n=\big\{x\in O : d(x,O^c)>1/n\big\}##
Prove that:
(a) ##O_n## is open and ##O_n\subset O## for all ##n\in \mathbb{N}##,
(b) ##O_1\subset O_2 \subset \dots ##, and ##\cup_n O_n=O##
(c) If ##O_n\neq 0## then ##d(O_n, O^c)\ge \frac{1}{n}##
(d) If ##O_n\neq 0## then ##d(O_n, O^c_{n+1})\ge \frac{1}{n(n+1)}##
Homework Equations
The Attempt at a Solution
(a) solved
(b) solved
(c) I'm not sure what the instructor is looking for here since there is no ##n## and no ##x\in O_n## such that ##d(x,O^c)=\frac{1}{n}## since that would contradict the construction of ##O_n##. It seems like the problem statement should be
If ##O_n\neq 0## then ##d(O_n, O^c)> \frac{1}{n}##.
(d) ##d(O_n,O^c_{n+1})=d(O_n,O^c)-d(O_{n+1},O^c)\ge \frac{1}{n}-\frac{1}{n+1}=\frac{1}{n(n+1)}##