Assume $(A_{i})_{i\in\Bbb N}$ to be an infinite sequence of sets of natural numbers, satisfying
$A_{0}\subseteq A_{1}\subseteq A_{2}\subseteq A_{3}\cdots\subseteq\Bbb N\tag{*}$
For each property $p_{i}$ shown below, state whether
• the hypothesis (*) is sufficient to conclude that $p_{i}$ holds; or
• the hypothesis (*) is sufficient to conclude that $p_{i}$ does not hold; or
• the hypothesis (*) is not sufficient to conclude anything about the truth of $p_{i}$ .
Justify your answers (briefly).
$p_{1}$ : $\forall k\in\Bbb N.\ A_{k}=\bigcup_{i=0}^{k}A_{i}$
$p_{2}$ : for all $i$, if $A_{i}$ is infinite, then $A_{i}=A_{i+1}$
$p_{3}$ : if $\forall i\in\Bbb N.\ A_{i}\neq A_{i+1}$, then $\bigcup_{i=0}^{\infty}A_{i}=\Bbb N$
$p_{4}$ : if $\forall i\in\Bbb N.\ A_{i}$ is finite, then $\bigcup_{i=0}^{\infty}A_{i}$ is finite
$p_{5}$ : if $\forall i\in\Bbb N.\ A_{i}$ is finite, then $\bigcup_{i=0}^{\infty}A_{i}$ is infinite
$p_{6}$ : if $\forall i\in\Bbb N.\ A_{i}$ is infinite, then $\bigcup_{i=0}^{\infty}A_{i}$ is infinite