Proving Limit of a Sequence: Simplest Method

[glebovg]

Homework Statement

What is the fastest way to prove this.
1/a_n→1/a, where a_n 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]

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.

 

[glebovg]

a_n is a sequence. I am trying to prove this limit law for the sequence.

 

[LCKurtz]

a_n 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.

 

[glebovg]

I just abriviated lim_n→∞ a_n = a as a_n→a (as n→∞) if that is what you mean.

 

[daveb]

Just to be clear, you're saying that if a_n goes to a, you want to prove (quickly) that 1/a_n goes to 1/a? Correct?

 

[LCKurtz]

Homework Statement

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

The attempt at a solution

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

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

I just abriviated lim_n→∞ a_n = a as a_n→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.

 

[glebovg]

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

 

[gb7nash]

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

You're stating some kind of conclusion. What's the hypothesis?

 

[LCKurtz]

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.

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

Did you even read my post?

 

[glebovg]

Let lim a_n = a, and lim b_n = b. Then, lim a_n/b_n = a/b.
We know lim (a_nb_n) = ab. So ...

 

[glebovg]

Never mind I found the neatest way to prove it.