Proving Limit of a Sequence: Simplest Method

In summary, the conversation is about proving the limit law for sequences, specifically 1/an→1/a, where an is a sequence. The conversation mentions the need for hypotheses and a correct statement of the theorem. The original statement of the theorem is deemed false, and a correct statement is given as: If lim an = a and lim bn = b, then lim (an/bn) = a/b. The conversation ends with the speaker claiming to have found a neat way to prove the theorem.
  • #1
glebovg
164
1
Homework Statement

What is the fastest way to prove this.
1/an→1/a, where an is a sequence.

The attempt at a solution

I know how to prove this but I am looking for a simple and elegant proof.
 
Physics news on Phys.org
  • #2
Of course, it isn't true the way you stated it. You need some hypotheses.

Elegant might mean noting that if a ≠ 0 then 1/x is continuous at a. Whether that is "simple" likely depends on the context.
 
  • #3
an is a sequence. I am trying to prove this limit law for the sequence.
 
  • #4
glebovg said:
an is a sequence. I am trying to prove this limit law for the sequence.

You haven't even stated the limit law correctly yet. And I thought you said you already know how to prove it. :confused:
 
  • #5
I just abriviated limn→∞ an = a as an→a (as n→∞) if that is what you mean.
 
  • #6
Just to be clear, you're saying that if an goes to a, you want to prove (quickly) that 1/an goes to 1/a? Correct?
 
  • #7
glebovg said:
Homework Statement

What is the fastest way to prove this.
1/an→1/a, where an is a sequence.

The attempt at a solution

I know how to prove this but I am looking for a simple and elegant proof.

LCKurtz said:
You haven't even stated the limit law correctly yet. And I thought you said you already know how to prove it. :confused:

glebovg said:
I just abriviated limn→∞ an = a as an→a (as n→∞) if that is what you mean.

No, I'm not talking about notation. I'm talking about the fact that you haven't stated the theorem correctly even yet. You need something in the form

If [hypotheses here] then [conclusion here].

Your original statement, highlighted above, not only doesn't do that, it is false.
 
  • #8
What do you mean it is false? How can a theorem be false? It has been proven. It is part of the Algebraic Limit Theorem.

1/an→1/a, where an is a sequence and a ≠ 0.
 
  • #9
glebovg said:
1/an→1/a, where an is a sequence and a ≠ 0.

You're stating some kind of conclusion. What's the hypothesis?
 
Last edited:
  • #10
LCKurtz said:
No, I'm not talking about notation. I'm talking about the fact that you haven't stated the theorem correctly even yet. You need something in the form

If [hypotheses here] then [conclusion here].

Your original statement, highlighted above, not only doesn't do that, it is false.

glebovg said:
What do you mean it is false? How can a theorem be false? It has been proven. It is part of the Algebraic Limit Theorem.

1/an→1/a, where an is a sequence and a ≠ 0.

Did you even read my post?
 
  • #11
Let lim an = a, and lim bn = b. Then, lim an/bn = a/b.
We know lim (anbn) = ab. So ...
 
  • #12
Never mind I found the neatest way to prove it.
 

FAQ: Proving Limit of a Sequence: Simplest Method

What is the simplest method for proving the limit of a sequence?

The simplest method for proving the limit of a sequence is to use the definition of a limit. This method involves showing that for any given epsilon (ε), there exists a natural number N such that for all n greater than N, the absolute value of the difference between the sequence and the limit is less than ε.

How is the definition of a limit used to prove the limit of a sequence?

The definition of a limit is used to prove the limit of a sequence by showing that as the terms of the sequence get closer and closer to the limit, the difference between the terms and the limit approaches zero. This is done by choosing an epsilon (ε) value and finding a natural number N that satisfies the definition for all n greater than N.

Can the simplest method be used for all sequences?

Yes, the simplest method can be used for all sequences as long as the sequence is convergent and the limit exists. However, for some more complex sequences, other methods such as the squeeze theorem or the monotone convergence theorem may be more efficient.

What are the advantages of using the simplest method for proving the limit of a sequence?

The main advantage of using the simplest method is that it is straightforward and easy to understand. It also provides a clear and logical approach to proving the limit of a sequence. Additionally, this method can be used for a wide range of sequences, making it a versatile tool for mathematicians.

Are there any limitations to using the simplest method for proving the limit of a sequence?

One limitation of using the simplest method is that it may not always provide the most efficient or elegant solution for proving the limit of a sequence. In some cases, there may be other methods that require less computation or provide a more concise proof. Additionally, this method may not be suitable for proving the limit of certain types of sequences, such as oscillating or divergent sequences.

Similar threads

Prove that the sequence does not have a convergent subsequence
Replies
4
Views
1K
How to show that these sequences are summable?
Replies
1
Views
1K
Proving convergence of rational sequence
Replies
2
Views
811
Show that a recursive sequence converges
Replies
4
Views
1K
Monotonic sequence definition of Continuity of a function
Replies
13
Views
1K
The Subsequential Limit Points of a Bounded Sequence
Replies
4
Views
1K
Proof for convergent sequences, limits, and closed sets?
Replies
11
Views
2K
Proof that a recursive sequence converges
Replies
5
Views
2K
What is the proof for the limit superior?
Replies
5
Views
2K
Prove: A bounded sequence contains a convergent subsequence.
Replies
6
Views
3K

Hot Threads

Recent Insights

Back
Top