# Thread: lim inf/sup innequality question...

1. ## lim inf/sup innequality question...

x_n and y_n are bounded

$\displaystyle \limsup x_n + \limsup y_n \geq \lim x_{r_n} + \lim y_{r_n}$

this is true because there presented sub sequences converge to the upper bound of x_n and y_n .
the sum of limits is the limit of sums so we get one limit
and the of two convergent sequence is one convergent sequence
$\displaystyle \lim (x_{r_n} + y_{r_n}) = \limsup (x_n+y_n)$

this is true because the sequence is constructed from a convergent to the sup sub sequences
so they equal the lim sup of $\displaystyle x_n+y_n$

now i need to prove that $\displaystyle \limsup (x_n+y_n) \Rightarrow \liminf x_n + \limsup y_n$

i tried: $\displaystyle \limsup (x_n+y_n) = \limsup x_n + \limsup y_n \Rightarrow \liminf x_n + \limsup y_n$
lim sup i always bigger then lim inf

so its true

did i solved it correctly?

2. Originally Posted by transgalactic
x_n and y_n are bounded
$\displaystyle lim sup x_n+lim sup y_n>=lim x_r_n+lim y_r_n$
this is true because there presented sub sequences converge to the upper bound of x_n and y_n .
the sum of limits is the limit of sums so we get one limit
and the of two convergent sequence is one convergent sequence
$\displaystyle lim(x_r_n+y_r_n)=limsup(x_n+y_n)$
this is true because the sequence is constructed from a convergent to the sup sub sequences
so they equal the lim sup of x_n+y_n

now i need to prove that
$\displaystyle limsup(x_n+y_n)=>liminf x_n+limsup y_n$

i tried:
$\displaystyle limsup(x_n+y_n)=limsup x_n+limsup y_n=>liminf x_n+limsup y_n$
lim sup i always bigger then lim inf

so its true

did i solved it correctly?

I'm not sure what you want to prove.
Is $\displaystyle r_n$ a particular subsequence?
And why do we know that there is a limit to $\displaystyle x_{r_n}$

3. x_r_n is a convergent subsequence of x_n

y_r_n is a convergent subsequence of y_n

r_n is just a sign to demonstrate that its a subsequence

by weirshrass laws to any bounded sequence there is a convergent subsequence

my proof is ok?

4. Originally Posted by transgalactic
x_n and y_n are bounded

$\displaystyle \limsup x_n + \limsup y_n \geq \lim x_{r_n} + \lim y_{r_n}$

this is true because there presented sub sequences converge to the upper bound of x_n and y_n .
the sum of limits is the limit of sums so we get one limit
and the of two convergent sequence is one convergent sequence
$\displaystyle \lim (x_{r_n} + y_{r_n}) = \limsup (x_n+y_n)$

this is true because the sequence is constructed from a convergent to the sup sub sequences
so they equal the lim sup of $\displaystyle x_n+y_n$

now i need to prove that $\displaystyle \limsup (x_n+y_n) \Rightarrow \liminf x_n + \limsup y_n$

i tried: $\displaystyle \limsup (x_n+y_n) = \limsup x_n + \limsup y_n \Rightarrow \liminf x_n + \limsup y_n$
lim sup i always bigger then lim inf

so its true

did i solved it correctly?

I'm still having trouble figuring out what you want to prove.
Also I don't think that
$\displaystyle \limsup (x_n+y_n) = \limsup x_n + \limsup y_n$.
What if $\displaystyle x_n=0,1,0,1,0,1,...$
and $\displaystyle y_n=1,0,1,0,1,0,...$, then
$\displaystyle x_n+y_n=1,1,1,1,1,1,...$, which has a limit and limit soupy of 1.
But the $\displaystyle \limsup x_n + \limsup y_n=1+1=2$.

If you just want to prove
$\displaystyle \limsup x_n + \limsup y_n \geq \lim x_{r_n} + \lim y_{r_n}$
well $\displaystyle \limsup x_n \geq \lim x_{r_n}$
and $\displaystyle \limsup y_n \geq \lim y_{r_n}$.
Now just add, but I'm still lost as to what you want.

5. $\displaystyle \lim (x_{r_n} + y_{r_n}) = \limsup (x_n+y_n)$

why its correct
what do i need to say so it will be valid
??

6. Originally Posted by transgalactic
$\displaystyle \lim (x_{r_n} + y_{r_n}) = \limsup (x_n+y_n)$

why its correct
what do i need to say so it will be valid
??

Let $\displaystyle w_n=x_n+y_n$.
Now I still need to understand this sequence $\displaystyle r_n$.
Is $\displaystyle r_n$ the sequence so that
$\displaystyle \limsup w_n=\lim w_{r_n}$?

7. r_n is not a sequence its just a way for me so sign a subsequence .

x_n is a bounded sequence with index n
and i want a subsequence to x_n
so i call it $\displaystyle x_{r_n}$ where n is the index again

its just a way to sign a subsequence

is it ok now?

8. can you show an example like this

there is such things on the internet