As another note, the way the concept was explained in class was that the limit L exists if I can always beat you at a game where you pick an then I pick an N. I win if given my N for every n > N,
I am having a lot of trouble with the concept of proving the limit of a sequence using epsilon n. As an example, I am working on trying to prove that the limit of the sequnce is . I understand that the first step is to set . I then solve for n and get . So I think I understand this tells me that for any , picking an will give a value less than . I'm just not sure where to go next with the proof. I've arbitraily tried different values than for the limit and every time I can solve arbitrary number for something with in it. So I must be doing something wrong as it seems using my method I can prove the limit to be any arbitrary number. In searching online and other forums it seems there is maybe a last step where I take my and somehow work in the "other direction" to prove that indeed any works, but I've been unable to follow. Thanks for any help.
Thanks for the quick reply! I fully understand everything right up to the last part.
This is the part that trips me up. so when I say I just plug that straight back into the right side of and go back to the beginning? I guess I'm have trouble understanding how that proves is the limit. For example, if I instead begin with I can get down to and also reverse my steps, but isn't the limit.So choose and reverse each step and you will have your proof
The reason it works is because you need to show there exists some value such that no matter how big it is, your function value is always within a band of width from the limit value. In symbols, that means if .