which of the following is true ?
$1.\sup_{0 <ε<1/2}\sup(A_{ε})<1$
$2.0<ε_{1}<ε_{2}<\frac{1}{2}\implies \inf(A_{ε_1})\lt \inf(A_{ε_2})$
$3.0<ε_{1}<ε_{2}<\frac{1}{2}\implies\sup(A_{ε_1})>\sup(A_{ε_2})$
$4.\sup(A_{ε_1}\bigcap\mathbb Q)=\sup\Bigl(A_{ε_1}\bigcap(\mathbb R$\ $\mathbb Q)\Bigr)$
I SEARChing this types of problem in Rudin ( analysis) book but I could not find this kind of problem and concept in any real anaylsis book.
so i very much confused about this problem
pliz help me
Q1-3) Give any $0\le\epsilon\le\frac{1}{2}, \sup A_\epsilon=\sup(0,1-\epsilon)=1-\epsilon$ and $\inf A_\epsilon=\sup(0,1-\epsilon)=0$ (Do you know why?)
Q4)Consider $A_\epsilon$, while $0\le\epsilon\le\frac{1}{2}$, forall $a<1-\epsilon$, exists $b\in \mathbb Q$ (Do you know why?) such that $a<b<1-\epsilon$, while $b$ can be taken to be inside $A_\epsilon$, that is, $b\in(A_\epsilon\bigcap\mathbb Q)$, since clearly $1-\epsilon$ is an upper bound of $(A_\epsilon\bigcap\mathbb Q),\sup(A_\epsilon\bigcap\mathbb Q)=1-\epsilon$, similarly, $\sup\Bigl(A_\epsilon\bigcap(\mathbb R\bigcap\mathbb Q)\Bigr)=1-\epsilon$.