Caroline Elliott

2022-01-23

Let $\left({a}_{n}\right)$, $\left({b}_{n}\right)$ be two bounded sequences.
a.Show that .
b. Give an example of $\left({a}_{n}\right),\left({b}_{n}\right)$ such that the strict inequality holds.

Hana Larsen

a)In view of $\left({a}_{n}\right)$ and $\left({b}_{n}\right)$ are bounded sequences,
$\underset{n\to \mathrm{\infty }}{lim}\supset max\left({a}_{n},{b}_{n}\right)\ge max\left(\underset{n\to \mathrm{\infty }}{lim}\supset {a}_{n},\underset{n\to \mathrm{\infty }}{lim}\supset {b}_{n}\right)$
We already know that lim sup is on the rise,
$⇒an\le max\left({a}_{n},{b}_{n}\right)$
(1)
Likewise,
$⇒{b}_{n}\le max\left({a}_{n},{b}_{n}\right)$
(2)
Thus,

b)Let's substitute $\left({a}_{n}\right)=\frac{1}{n},\left({b}_{n}\right)=\frac{1}{2n}$
we know that $\left({a}_{n}\right)$ and $\left({b}_{n}\right)$ are bounded,
$⇒max\left({a}_{n},{b}_{n}\right)=max\left(\frac{1}{n},\frac{1}{2n}\right)$
$=max\left\{\left(1,12,13,\dots \right),\left(12,14,\dots \right)\right\}$
(3)
However,
$\underset{n\to \mathrm{\infty }}{lim}\supset \left({a}_{n}\right)=0$ and $\underset{n\to \mathrm{\infty }}{lim}\supset \left({b}_{n}\right)=0$
$max\left(\underset{n\to \mathrm{\infty }}{lim}\supset {a}_{n},\underset{n\to \mathrm{\infty }}{lim}\supset {b}_{n}\right)=0$ (4)
Then by the equation (3) and equation (4):

Therefore the given is showed and the example of $\left({a}_{n}\right)$,

Do you have a similar question?