Simple notation question about subsequences

[gottfried]

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

 

[Office_Shredder]

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

You have some notation confusion. The correct comparison is a_nm with a_m. The n_m 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, n_m ≥ m for all m, so if your sequence is decreasing you would get a_nm ≤ a_m

 

[Fredrik]

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.

 

[gottfried]

Thanks guys.