Analysis Problem, limits & supremum, infimum and sequences

In summary, The conversation discusses a proof involving bounded sequences in R and the limit inferior. It is shown that the limit inferior of the sum of two sequences is greater than or equal to the sum of the limit inferiors of each sequence. One of the participants had overlooked a solved example but was able to solve the problem by using the result from a previous problem.
  • #1
36
0
I have analysis quiz tomorrow and i am really poor at sequences.

I don't know where to begin

Let (sn) and (tn) be sequences in R. Assume that (sn) is bounded.

Prove that liminf(sn +tn)≥liminfsn +liminftn,

where we define −∞ + s = −∞ and +∞ + s = +∞ for any s ∈ R.

-thanks
 
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  • #2
Ok, what did you try already?
 
  • #3
I got it,

I had to solve a problem before this one which gave me the result
lim infSn = -lim sup(-Sn)

And from a solved example i got

lim Sup(sn + tn) < lim supSn + lim SupTn

so multiplying b.s by (-1) and using -Sn instead for
we get

-lim Sup(-Sn - T) > lim Sup(-Sn) + lim Sup(-Tn)

There fore

lim Inf(Sn + Tn) > lim InfSn + lim InfTn

I had overlooked the solved example.
 

1. What is an analysis problem?

An analysis problem is a mathematical problem that involves using various methods and techniques to study and understand a given set of data or mathematical objects. It often requires breaking down complex systems or ideas into smaller, more manageable parts in order to solve them.

2. What are limits in mathematics?

Limits in mathematics refer to the value that a function or sequence approaches as its input or index approaches a specific value. It is a fundamental concept in calculus and is used to describe the behavior of functions near a specific point or as the input value increases or decreases infinitely.

3. What is the supremum of a set?

The supremum of a set is the smallest upper bound of that set. In other words, it is the highest value that a set can approach without ever exceeding it. It is often denoted as sup(A) and is an important concept in mathematical analysis and optimization problems.

4. What is the infimum of a set?

The infimum of a set is the largest lower bound of that set. It is the lowest value that a set can approach without going below it. It is also denoted as inf(A) and is commonly used in mathematical analysis to find the minimum or maximum of a function.

5. What are sequences in mathematics?

In mathematics, a sequence is a list of numbers or objects that are arranged in a specific order. Each element in the sequence is called a term, and the position of each term is determined by its index. Sequences are important in many areas of mathematics, such as number theory, calculus, and statistics.

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