# Partial limit

## Homework Statement

I have to find all the partial limits {I hope this is how this term named in English} of a sequences

## Homework Equations

$$a_1=0$$

$$a_{2n}=\frac {a_{2n-1}} {3}$$

$$a_{2n+1} = 1/3 + a_{2n}$$

## The Attempt at a Solution

I have tried to prove first that sequences of all the even terms converges due to fact that sequence is monotonic and have a suprimum, but have failed to prove it.
Another problem is that subsequences of odd term is non monotonic, but I also can't use the Cantor's Lemma.

Could you please suggest how to approach this problem?
Thanks.

Last edited:

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Mark44
Mentor

## Homework Statement

I have to find all the partial limits {I hope this is how this term named in English} of a sequences

## Homework Equations

$$a_1=0$$

$$a_{2n}=\frac {a_{2n-1}} {3}$$

$$a_{2n+1} = 1/3 + 2_{2n}$$
In the equation above do you mean a2n+1 = 1/3 + 22n?

## The Attempt at a Solution

I have tried to prove first that sequences of all the even terms converges due to fact that sequence is monotonic and have a suprimum, but have failed to prove it.
Another problem is that subsequences of odd term is non monotonic, but I also can't use the Cantor's Lemma.

Could you please suggest how to approach this problem?
Thanks.

Sorry, have fixed it in my first post.

Mark44
Mentor
You're looking at the two subsequences: one with the odd-index terms and the other with the even-index terms. Have you calculated the first dozen or so terms of your sequence?

I've calculated again some terms of the sequence and found out that I did a mistake in my previous calculation as both subsequences seem to be monotonic, but I can't find a way to prove that the sequences have suprimums.

0, 0, 81/243, 27/243, 108/243, 36/243, 36/243, 117/243, 39/243, 120/243, 40/243, 121/243

I have figured it out, thanks.