Picking sequence from a given set

[wayneckm]

Hello all,

I come across this question while reading a proof.

Given a set $$A$$ with some property related to $$f$$ which is a function and an element $$b$$, they may begin with, in the proof, choosing a sequence $$a_{n} \in A$$ such that $$f(a_{n} b) \rightarrow f(b)$$, then they keep going until they prove the limit $$a_{\infty}$$ exists and is also inside $$A$$.

So how and why can they start with extracting a sequence $$a_{n}$$ with the desired limiting property, $$f(a_{n} b) \rightarrow f(b)$$? I am wondering how and why they can be sure such kind of sequence can be found? Or indeed under what kind of conditions (closure or what else?) this is guaranteed to be true? Apparently this kind of argument/technique appears quite often in some proofs.

Thanks.


Wayne

 

[jvicens]

, thank you for your question. The extraction of a sequence a_{n} with the desired limiting property, f(a_{n} b) \rightarrow f(b) , is a common technique in mathematical proofs. It is based on the concept of convergence, which is a fundamental concept in analysis.

In order to understand why this technique is used, it is important to understand the definition of convergence. A sequence a_{n} is said to converge to a limit a_{\infty} if for any positive real number \epsilon , there exists a natural number N such that for all n \geq N , the distance between a_{n} and a_{\infty} is less than \epsilon . In other words, as the terms in the sequence get closer and closer to the limit, the distance between them becomes smaller and smaller.

Now, in the context of your question, we have a set A and a function f that is related to some property of the set. In order to prove that the limit a_{\infty} exists and is also in A , we need to show that the sequence a_{n} converges to a_{\infty} and that a_{\infty} is in A . This is where the technique of extracting a sequence with the desired limiting property comes in.

By choosing a sequence a_{n} \in A such that f(a_{n} b) \rightarrow f(b) , we are essentially ensuring that the terms in the sequence are getting closer and closer to the limit a_{\infty} . This is because we know that f(a_{n} b) is related to the property of the set A , and therefore, as a_{n} gets closer to a_{\infty} , f(a_{n} b) will also get closer to f(b) .

In terms of conditions, this technique is typically used when the function f is continuous. This means that small changes in the input of the function result in small changes in the output. In this case, as the terms in the sequence a_{n} get closer to the limit a_{\infty} , the corresponding values of f(a_{n} b)