Limit superior/inferior

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In summary, the properties of lim sup (x+y) < & = lim sup x + lim sup y and lim sup (xy) < & = lim sup x * lim sup y can be proven using a lemma that states if a sequence has a supremum, then there exists a subsequence that converges to that supremum. By applying this lemma, it can be shown that the supremum of the sum of two sequences is less than or equal to the sum of the supremums of each individual sequence. However, there may be some issues with terminology when defining the subsequence.
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Hello, i just started in real analysis not too long ago. In my lecture notes There are 2 properties
lim sup (x+y) < & = lim sup x + lim sup y
lim sup (xy) < & = lim sup x * lim sup y
however there are no proofs for them in my notes. so i was wondering if anyone knew where a proof of these properties might lay or possibily give me a proof for these.
Thanks
 
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  • #2
Take a following lemma.

If [tex]a=\textrm{sup}\{a_1,a_2,a_3,...\}[/tex], then there exists a subsequence [tex]a_{k_1},a_{k_2},...[/tex] so that [tex]\lim_{i\to\infty} a_{k_i} = a[/tex].

Using this it is possible to conclude that

[tex]\textrm{sup}\{x_n+y_n, x_{n+1}+y_{n+1},...\} \leq \textrm{sup}\{x_n,x_{n+1},...\} + \textrm{sup}\{y_n,y_{n+1},...\}[/tex]

and it's almost done.

EDIT: I used bad terminology. I guess for example [tex]a_1,a_1,a_1,...[/tex] isn't really a subsequence. Anyway, now that should be accepted as a "subsequence". The idea is then to take such a subsequence from the sequence of sums (on the left side of the inequality).
 
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  • #3


The properties you mentioned are known as the limit superior and limit inferior (also known as the lim sup and lim inf) of a sequence. They are important concepts in real analysis, and it is great that you are already starting to learn about them.

To prove these properties, we first need to understand what the limit superior and limit inferior are. The limit superior of a sequence (x_n) is defined as the largest number that the sequence approaches as n tends to infinity. In other words, it is the supremum (or least upper bound) of all the numbers in the sequence. Similarly, the limit inferior is the infimum (or greatest lower bound) of the sequence.

Now, let's look at the first property: lim sup (x+y) < & = lim sup x + lim sup y. This property states that the limit superior of the sum of two sequences is less than or equal to the sum of their individual limit superiors. To prove this, we can use the definition of limit superior. Let's denote the limit superior of the sequence (x_n) as S and the limit superior of the sequence (y_n) as T. This means that for any ε > 0, there exists an N_1 such that for all n > N_1, we have x_n < S + ε. Similarly, for the sequence (y_n), there exists an N_2 such that for all n > N_2, we have y_n < T + ε.

Now, let's consider the sequence (x_n + y_n). For any ε > 0, we can choose N = max{N_1, N_2}. Then for all n > N, we have x_n + y_n < S + T + 2ε. This means that S + T + 2ε is an upper bound for the sequence (x_n + y_n). But since S is the supremum of the sequence (x_n), we know that x_n < S + ε for all n > N_1. Therefore, x_n + y_n < S + T + 2ε for all n > N_1. This shows that S + T + 2ε is also an upper bound for the sequence (x_n + y_n). Since ε is arbitrary, we can conclude that S + T is the supremum of the sequence (x_n + y_n), which is the definition of lim sup (
 

1. What is the definition of limit superior/inferior?

The limit superior/inferior of a sequence is the highest/lowest number that the sequence approaches as the number of terms approaches infinity.

2. How is limit superior/inferior different from the regular limit?

The regular limit looks at the behavior of a sequence at a specific point, while the limit superior/inferior looks at the overall behavior of the sequence as the number of terms approaches infinity.

3. How is limit superior/inferior used in real-world applications?

Limit superior/inferior can be used in various fields such as economics, physics, and engineering to analyze the behavior of systems over time.

4. How is limit superior/inferior calculated?

Limit superior/inferior can be calculated by finding the maximum/minimum of the set of all accumulation points of the sequence.

5. What is the importance of understanding limit superior/inferior?

Understanding limit superior/inferior can help in predicting the long-term behavior of a system or sequence and can provide insights into its overall behavior.

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