Limits Question: Proving Sn <= b for Finite n

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In summary, the conversation discusses a proof by contradiction to show that if a sequence converges and is always less than or equal to a constant for all but finitely many terms, then the limit of the sequence is also less than or equal to that constant. The conversation also briefly mentions using latex for the proof and considers the implication of |s_n - s| < \epsilon = s-b for all n > N.
  • #1
phygiks
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Hello,
I'm having trouble with this question.
Let Sn be a sequence that converges. Show that if Sn <= b for all but finitely many n, then lim sn <= b.
This is what I'm trying to do, assume s = lim Sn and s > b. (Proof by contradiction) abs(Sn-s) < E, E > 0. Don't know what to do from there, but maybe set E = s -b. E is epsilon by the way. Probably to start using latex...

If anyone could help, that would be awesome.
 
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  • #2
Your idea is fine so far. Now what does [tex]|s_n - s| < \epsilon = s-b[/tex] for all n > N imply?
 
  • #3
s is a upper bound, so the Sn-s is negative. So abs(Sn-s) < s -b doesn't hold true all n. I'm not sure though
 

Related to Limits Question: Proving Sn <= b for Finite n

1. What is the definition of a limit?

A limit is a mathematical concept that describes the behavior of a function as its input approaches a certain value. It is denoted by the notation "lim" and can be used to determine the value of a function at a given point or to analyze the behavior of a function at infinity.

2. How do you prove Sn <= b for finite n?

To prove Sn <= b for finite n, you can use the definition of a limit and the properties of limits. This involves showing that for any given epsilon (ε) greater than 0, there exists a corresponding natural number N such that for all n ≥ N, Sn <= b. This essentially means that the terms of the sequence Sn get closer and closer to the limit b as n increases.

3. Can you give an example of a limit question involving Sn <= b for finite n?

One example of a limit question involving Sn <= b for finite n is: Prove that the limit of 1/n as n approaches infinity is equal to 0. This can be shown by choosing any ε > 0 and finding a corresponding N such that 1/n < ε for all n ≥ N. Therefore, the limit of 1/n as n approaches infinity is 0.

4. What are some common techniques for proving Sn <= b for finite n?

Some common techniques for proving Sn <= b for finite n include using the definition of a limit, using the squeeze theorem, and using algebraic manipulations to simplify the expression. It is also important to understand the properties of limits, such as the limit laws and the sandwich theorem.

5. How is the concept of a limit used in real-world applications?

The concept of a limit is used in various real-world applications, such as in physics, engineering, and economics. For example, in physics, limits are used to describe the velocity and acceleration of an object as it moves towards a certain point. In economics, limits are used to analyze the behavior of supply and demand functions as they approach certain values. In general, limits help us understand the behavior of a system as it approaches a specific point or value.

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