Let (an), (bn) be two bounded sequences. a.Show that limn→∞⊃ max(an,bn)≥max(limn→∞⊃an,limn→∞⊃bn). b. Give an example...

a)In view of ${\displaystyle \left(a_n\right)}$ and ${\displaystyle \left(b_n\right)}$ are bounded sequences,
${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\supset max(a_n,b_n)\ge max(\underset{n\to \mathrm{\infty }}{lim}\supset a_n,\underset{n\to \mathrm{\infty }}{lim}\supset b_n)}$
We already know that lim sup is on the rise,
${\displaystyle \Rightarrow an\le max(a_n,b_n)}$
${\displaystyle \Rightarrow lim\supset max(a_n,b_n)\ge lim\supset a_n}$ (1)
Likewise,
${\displaystyle \Rightarrow b_n\le max(a_n,b_n)}$
${\displaystyle \Rightarrow lim\supset max(a_n,b_n)\ge lim\supset b_n}$ (2)
Thus,
${\displaystyle lim\supset max(a_n,b_n)\ge max(lim\supset a_n,lim\supset b_n)}$
b)Let's substitute ${\displaystyle \left(a_n\right)=\frac{1}{n},\left(b_n\right)=\frac{1}{2n}}$
we know that ${\displaystyle \left(a_n\right)}$ and ${\displaystyle \left(b_n\right)}$ are bounded,
${\displaystyle \Rightarrow max(a_n,b_n)=max(\frac{1}{n},\frac{1}{2n})}$
${\displaystyle =max\{(1,12,13,\dots ),(12,14,\dots )\}}$
${\displaystyle lim\supset max(a_n,b_n)=1}$ (3)
However,
${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\supset \left(a_n\right)=0}$ and ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\supset \left(b_n\right)=0}$
${\displaystyle max(\underset{n\to \mathrm{\infty }}{lim}\supset a_n,\underset{n\to \mathrm{\infty }}{lim}\supset b_n)=0}$ (4)
Then by the equation (3) and equation (4):
${\displaystyle lim\supset max(a_n,b_n)>max(lim\supset a_n,lim\supset b_n)}$
Therefore the given ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\supset max(a_n,b_n)\ge max(\underset{n\to \mathrm{\infty }}{lim}\supset a_n,\underset{n\to \mathrm{\infty }}{lim}\supset b_n)}$ is showed and the example of ${\displaystyle \left(a_n\right)}$,