Lim inf/sup innequality question

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In summary, this conversation discusses the boundedness of sequences and the relationship between the lim sup and lim inf of the sums of two sequences. By using the properties of lim sup and lim inf, it is shown that lim sup(x_n+y_n) is equal to limsup x_n+limsup y_n, and that limsup(x_n+y_n) is always bigger than liminf x_n+limsup y_n. This proves that limsup(x_n+y_n) >= liminf x_n+limsup y_n.
  • #1
transgalactic
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x_n and y_n are bounded
[tex]
lim sup x_n+lim sup y_n>=lim x_r_n+lim y_r_n

[/tex]
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
[tex]
lim(x_r_n+y_r_n)=limsup(x_n+y_n)
[/tex]
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
[tex]
limsup(x_n+y_n)=>liminf x_n+limsup y_n
[/tex]

i tried:
[tex]
limsup(x_n+y_n)=limsup x_n+limsup y_n=>liminf x_n+limsup y_n
[/tex]
lim sup i always bigger then lim inf

so its true

did i solved it correctly?
 
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  • #2


Yes, your solution is correct. By using the properties of lim sup and lim inf, you have shown that lim sup(x_n+y_n) is equal to limsup x_n+limsup y_n, and since lim sup is always bigger than lim inf, it follows that limsup(x_n+y_n) is also bigger than liminf x_n+limsup y_n. This proves that limsup(x_n+y_n) >= liminf x_n+limsup y_n. Great job!
 

1. What is the definition of lim inf/sup inequality?

Lim inf/sup inequality is a mathematical concept that states that the limit inferior of a sequence of numbers is less than or equal to the limit superior of the same sequence.

2. How is lim inf/sup inequality used in real-world applications?

Lim inf/sup inequality is commonly used in the analysis of sequences and series in mathematics, but it also has applications in physics, engineering, and computer science.

3. What is the difference between lim inf/sup inequality and strict inequality?

In strict inequality, the two values being compared must be unequal. In lim inf/sup inequality, the two values can be equal, but the lim inf must still be less than or equal to the lim sup.

4. Can lim inf/sup inequality be applied to infinite sequences?

Yes, lim inf/sup inequality can be applied to infinite sequences as long as the sequence is well-defined and converges.

5. What happens if lim inf/sup inequality is violated?

If lim inf/sup inequality is violated, it means that the sequence being analyzed is not well-behaved and may not converge or have a well-defined limit.

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