Hello everyone. I am posting this not because the normal proof is difficult, but because I think I have come up with an alternative way of doing it and am wondering if I made any fatal errors. Also, half the reason I am putting this up here is to improve my often remarked upon lack of coherency. So any constructive criticism in either category is much appreciated.
Definitions: If is a metric space and is a sequence under , then we define an infinite subsequence of as follows: If we have a infinite sequence of natural numbers where then the sequence is a subsequence of .
So now that we have defined an infinite subsequence, let's get to the question.
Question: If is a convergent sequence under which converges to then prove that every infinite subsequence of converges to .
Answer: I think I know how they intended it to be done, but I came up with a different method and would appreciate any crticism.
1. In the definition of the subsequence we defined the in to be sequence of natural numbers such that . So instead of let us define it as follows: let be represented by which is a function of and has the quality that it is monotonically increasing, so then we may represent a subsequence of as .
2. We know because converges that for every there exists and integer such that .
3. So from this we can make the observation that whenever . Where is the generalized inverse function.
4. Stating the end part of 4. more explicitly: for every there exists an integer such that whenever . This the definition of a sequence which converges to the value which is what was asked to be proven.
Any comments greatly appreciated,