# Homework Help: Limit of a Sequence

1. Oct 4, 2011

### glebovg

The problem statement, all variables and given/known data

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.

2. Oct 4, 2011

### 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.

3. Oct 5, 2011

### glebovg

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

4. Oct 5, 2011

### LCKurtz

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

5. Oct 5, 2011

### glebovg

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

6. Oct 5, 2011

### daveb

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. Oct 5, 2011

### 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.

8. Oct 5, 2011

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

9. Oct 5, 2011

### gb7nash

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

Last edited: Oct 5, 2011
10. Oct 5, 2011

### LCKurtz

Did you even read my post?

11. Oct 5, 2011

### glebovg

Let lim an = a, and lim bn = b. Then, lim an/bn = a/b.
We know lim (anbn) = ab. So ...

12. Oct 5, 2011

### glebovg

Never mind I found the neatest way to prove it.