# Simple notation question about subsequences

1. May 7, 2013

### gottfried

Hey guys,
I was just reading notes on metric spaces and was wondering if the notation of subsequences was such that anm is always further down the original sequence than an? For example suppose you have a strictly decreasing sequnce does the notation imply that anm<an

2. May 7, 2013

### Office_Shredder

Staff Emeritus

You have some notation confusion. The correct comparison is anm with am. The nm is supposed to be a sequence of integers n1, n2, etc. and you don't get to specify what n is, you specify what m is. But your intuition is correct, nm ≥ m for all m, so if your sequence is decreasing you would get anm ≤ am

3. May 7, 2013

### Fredrik

Staff Emeritus
The question doesn't make sense as it's written, but you probably meant $a_m$ when you said $a_n$. So the question is (I think) if $n_m\geq m$ for all m.

The answer to that is yes, the map $m\mapsto n_m$ is increasing. If it wasn't, we wouldn't be dealing with a subsequence, but a rearrangement of a subsequence.

I'm moving the post to general math. Since it's not a question about a textbook-style problem, it doesn't belong in homework, and since you're not asking about convergence, it doesn't really belong in topology & analysis either.

Edit: I didn't see Office_Shredder's post until after I had finished mine.

4. May 7, 2013

Thanks guys.