Proving A_n Converges to 0: Real Analysis

In summary, the problem is to prove that the sequence A_n = (a, a, a, a, ...) converges to a, where a is a vector in R^p. The confusion arises from thinking that a is a point in R^p, when in fact it is a vector. Using the distance function, it can be shown that the sequence converges to a for any given epsilon.
  • #1
junior33
3
0
prove [tex] \ A_n = ( a, a,a,a,a,...) [/tex] converges to zero. [tex] a \in \ R^p [/tex]

Been reading this real analysis book before i take it next semester and been a lil stuck on this question. I am probably making it seem more difficult than it is. Most of the questions had examples in the chapter but this one didnt. can some one help me out?
 
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  • #2
If A_n = (1_1,1_2,...1_p) then A_n is constantly point with coordinates 1. This An will NOT converge to 1, it will converge to the constant itself.

i think you are confusing a point in R^p with the sequence itself a point in R^p is a set of p real numbers where the order in which each of these numbers follow each other matters.
a itself is NOT a point in R^p.
 
  • #3
it says that the a's are vectors in [tex] a \in \ R^p [/tex]

would it be that same
 
  • #4
sorry, that's correct. the a's ARE vectors. sorry i thought you thought they were coordinates of the vectors in the sequence.

well use the distance function u have and choose N=1 for any epsilon. see what happens.
 
  • #5
First, as the sticky at the top of this section says, this is NOT the place for homework. I am moving it to the homework section.

Second, you misstated the problem in the body of your post. You do NOT want to prove "that (a, a, a, a, ...) converges to 0" because, in general, it doesn't. You want to prove that it converges to a. Okay what is |a- a|?
 
  • #6
^^^ yes that's what i meant
 

FAQ: Proving A_n Converges to 0: Real Analysis

What is the definition of convergence in real analysis?

The definition of convergence in real analysis is that a sequence of real numbers, {a_n}, converges to a real number L if for any positive real number ε, there exists a natural number N such that for all n ≥ N, the absolute value of (a_n - L) is less than ε.

How do you prove that a sequence converges to 0?

To prove that a sequence, {a_n}, converges to 0, you must show that for any positive real number ε, there exists a natural number N such that for all n ≥ N, the absolute value of a_n is less than ε.

What is the significance of proving that A_n converges to 0?

Proving that A_n converges to 0 is significant because it shows that the terms of the sequence are getting closer and closer to 0, meaning that the sequence is approaching a limit. This is important in many areas of mathematics and science, as it allows for the prediction and analysis of different phenomena.

What are some common techniques used in proving that A_n converges to 0?

Some common techniques used in proving that A_n converges to 0 include the epsilon-delta definition of convergence, the squeeze theorem, and the Cauchy criterion. These techniques involve manipulating the terms of the sequence and using mathematical properties to show that they approach 0.

Can a sequence converge to 0 in more than one way?

Yes, a sequence can converge to 0 in more than one way. For example, a sequence may converge to 0 at a constant rate, or it may approach 0 in a more erratic or oscillating manner. Different sequences may also require different proof techniques to show convergence to 0.

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