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Vector valued functions

Prove that lim⁡t→pF(t)=A⟺lim⁡t→p∥F(t)−A∥=0\lim\limits_{t \to p}F(t) = A \Longleftrightarrow \lim\limits_{t \to p} \parallel F(t)-A \parallel = 0


I can just write the standard definition first.

Let F(t)=(f1(t)⋯fn(t))F(t)=(f_1(t)\cdots f_n(t)) and A=(a1⋯an)A=(a_1\cdots a_n)

Then lim⁡t→pF(t)=A⟺lim⁡t→p(f1(t)⋯fn(t))=(a1⋯an)\lim\limits_{t \to p}F(t)=A \Longleftrightarrow \lim\limits_{t \to p}(f_1(t)\cdots f_n(t))=(a_1\cdots a_n)

Then how should I proceed? :colondollar:
Original post by AishaGirl
Prove that lim⁡t→pF(t)=A⟺lim⁡t→p∥F(t)−A∥=0\lim\limits_{t \to p}F(t) = A \Longleftrightarrow \lim\limits_{t \to p} \parallel F(t)-A \parallel = 0


I can just write the standard definition first.

Let F(t)=(f1(t)⋯fn(t))F(t)=(f_1(t)\cdots f_n(t)) and A=(a1⋯an)A=(a_1\cdots a_n)

Then lim⁡t→pF(t)=A⟺lim⁡t→p(f1(t)⋯fn(t))=(a1⋯an)\lim\limits_{t \to p}F(t)=A \Longleftrightarrow \lim\limits_{t \to p}(f_1(t)\cdots f_n(t))=(a_1\cdots a_n)

Then how should I proceed? :colondollar:


Not something I'm experienced with here, so this might be a wild stab, but can't you just prove this using your standard real analysis definition of tending to a finite limit??
Original post by RDKGames
Not something I'm experienced with here, so this might be a wild stab, but can't you just prove this using your standard real analysis definition of tending to a finite limit??


But is that actually proving it?


If I do something like (lim⁡t→p∣f1(t)−a∣⋯lim⁡t→p∣fn(t)−a∣)=(0⋯0)⟺lim⁡t→p∣F(t)−A∣=0\Big(\lim\limits_{t \to p} |f_1(t)-a | \cdots \lim\limits_{t \to p} |f_n(t)-a |\Big)=(0\cdots 0) \Longleftrightarrow \lim\limits_{t \to p} |F(t)-A|=0


But have I actually proven it or just rewritten the original statement? I feel like I have not proven it.
(edited 7 years ago)
Original post by AishaGirl
Prove that lim⁡t→pF(t)=A⟺lim⁡t→p∥F(t)−A∥=0\lim\limits_{t \to p}F(t) = A \Longleftrightarrow \lim\limits_{t \to p} \parallel F(t)-A \parallel = 0

I think you need to give us some more information here. What you have written is to my mind the definition of convergence of vectors in a Euclidean space, so no proof is possible.

In what context has this question arisen?
Original post by atsruser
I think you need to give us some more information here. What you have written is to my mind the definition of convergence of vectors in a Euclidean space, so no proof is possible.

In what context has this question arisen?


There is no more information, that's it. It just says "prove that..." and that's it.
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Zacken
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Original post by AishaGirl
There is no more information, that's it. It just says "prove that..." and that's it.


What's your definition of convergence for vectors?
Original post by AishaGirl
There is no more information, that's it. It just says "prove that..." and that's it.


Well, I suspect that there must be more. For example:

1. In what course has this come up?
2. What kind of vector spaces are you working with? How is the norm defined?
3. How have you defined limits of vectors?
4. What do you already know about limits?
Attached is my interpretation of the problem. Throughout my solution I refer to basic limit properties from the website
http://tutorial.math.lamar.edu/Classes/CalcI/LimitProofs.aspx#Extras_Limit_LimitProp
vector_valued.pdf85.8KB

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