Proving Properties of Lim Sup: Sequences and Limits

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In summary, the limit superior of a sequence is the limit of the supremum of the sequence as n approaches infinity. To prove (iii), it follows from the fact that the supremum of a constant multiplied by a sequence is equal to the constant multiplied by the supremum of the sequence. For (i) and (ii), we need to prove that for any given epsilon and N, there exists a k that satisfies the given conditions. To prove (i), we can assume it is false and use the given conditions to find a contradiction. To prove (ii), we can use the given conditions to get an upper bound for lim sup that is less than L, leading to a contradiction.
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Bashyboy
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Homework Statement



The limit superior of a sequence ##(a_n)## is defined as ##\overline{\lim}_{n \to \infty} a_n = \lim_{n \to \infty} \sup \{a_k ~|~ k \ge n \}##. Letting ##L = \overline{\lim}_{n \to \infty} a_n##, I am asked to prove the following:

(i) For each ##N## and for each ##\epsilon > 0##, there exists some ##k > N## such that ##a_k \ge L - \epsilon##.

(ii) For each ##\epsilon > 0##, there is some ##N## such that ##a_k \le L + \epsilon## for all ##k > N##.

(iii) ##\overline{\lim}_{n \to \infty} ca_n = c \overline{\lim}_{n \to \infty} a_n##

Homework Equations

The Attempt at a Solution



(iii) is rather easy to prove: it just follows from the fact that ##\sup(cA) = c \sup (A)## and ##\lim_{n \to \infty} c x_n = c \lim_{n \to \infty} x_n## for ##c \ge 0##, so I will move on to the first and second part

Let ##\epsilon > 0## and ##N \in \Bbb{N}##. Since ##L = \overline{\lim}_{n \to \infty} a_n##, then there exists a ##K \in \Bbb{N}## such that ##| \sup \{a_k ~|~ k \ge n \} - L | < \epsilon## for every ##n \ge K## or ##L - \epsilon < \sup \{a_k ~|~ k \ge n \} < \epsilon + L## for every ##n \ge K##.. On the one hand, we get ##a_k \le \sup \{a_k ~|~ k \ge n \} < \epsilon + L## for every ##k \ge K##, which proves part (ii). However, I don't see how to prove part (i). Also, as you may have noticed, I can only get strict inequality in part (ii).
 
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To prove part (i), assume it is false, so that there is some ##\epsilon>0## and positive integer ##N## such that ##\forall k>N:\ a_k<L-\epsilon##.

Can you get an upper bound for lim sup of the sequence from that, that is less than ##L##? If so, that's the contradiction we need.
 

Related to Proving Properties of Lim Sup: Sequences and Limits

What is Lim Sup?

Lim Sup, short for limit superior, is a concept in mathematical analysis that is used to describe the behavior of a sequence of numbers or functions as the index approaches infinity. It is defined as the smallest number that is greater than or equal to all of the elements in the sequence.

How is Lim Sup calculated?

In order to calculate Lim Sup, one must first have a sequence of numbers or functions. Then, the Lim Sup is found by taking the supremum (or least upper bound) of the set of all the numbers or values in the sequence. In other words, it is the maximum limit of the sequence.

What is the relationship between Lim Sup and Lim Inf?

Lim Sup and Lim Inf (limit inferior) are two concepts that are closely related and often used together. While Lim Sup describes the maximum limit of a sequence, Lim Inf describes the minimum limit. In other words, Lim Sup is the upper bound and Lim Inf is the lower bound of a sequence.

What are the properties of Lim Sup?

There are several key properties of Lim Sup that are important to understand. These include: 1) Lim Sup is always greater than or equal to the elements in the sequence; 2) Lim Sup is the smallest number that is greater than or equal to all of the elements in the sequence; 3) Lim Sup is unique for a given sequence; and 4) Lim Sup is a monotonic decreasing function.

What is the significance of Lim Sup in mathematics and science?

Lim Sup is a fundamental concept in mathematics and is used in various fields such as analysis, calculus, and probability theory. It is also important in the study of limits, continuity, and convergence of sequences and functions. In science, Lim Sup is used to describe the behavior of physical systems and to analyze data in various fields such as physics, biology, and economics.

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