Absolute Value of Limits Proof

In summary, the conversation discusses how to prove that if a sequence bn converges to b, then the sequence of absolute values |bn| also converges to |b|. The conversation also considers the converse of this statement and concludes that it is not true in all cases. It suggests starting by writing down the assumption and what needs to be proven formally to make the proof more clear.
  • #1
tnocel1
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Homework Statement



Show that if bn→b, then the sequence of absolute values |bn| converges to |b|.

Homework Equations





The Attempt at a Solution



I've been proving various properties of limits, including product of limits and sum of products, but have been having trouble making progress with the approach to absolute value of limits. I was also wondering is the converse of this is true, that is if |bn|→|b|, is it also true that (bn)→b
 
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  • #2
Not at all, for example |(-1)^n| -> 1, but (-1)^n doesn't converge at all!
How about you start by writing down formally the assumption and what you want to prove. You will see that it's pretty straight forward then.
 

1. What is the definition of absolute value of limits?

The absolute value of limits is a mathematical concept that measures the distance between a function and a specific value as the input approaches that value. It is denoted by the symbol "|" and is used to ensure that the resulting value is always positive.

2. How do you prove the absolute value of limits?

To prove the absolute value of limits, we can use the definition of limit and apply it to the function in question. We then manipulate the limit expression to show that it is equal to the absolute value of the limit.

3. Can the absolute value of limits be negative?

No, the absolute value of limits can never be negative. This is because the absolute value function always returns a positive value, regardless of the sign of the input. Therefore, the absolute value of limits will always be positive.

4. What is the purpose of the absolute value of limits?

The absolute value of limits is used to determine the behavior of a function as the input approaches a specific value. It helps us understand how the function behaves near that value and can be used to evaluate the continuity and differentiability of a function at a point.

5. How is the absolute value of limits used in real-life applications?

The absolute value of limits is used in various fields such as engineering, physics, and economics to model and analyze real-world phenomena. For example, in physics, it is used to determine the velocity and acceleration of an object at a specific point in time. In economics, it is used to calculate the marginal cost and revenue of a product.

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