I have just started learning elementary analysis and am trying to understand exactly what the precise definition of a sequence to converge means, despite doing well on the problems in the book, including the more abstract, general proofs, but stilll don't really understand exactly what is going on here. So the definition is:
if and only if for any , there exists such that implies that .
I was told to think of it as a challenge. What do they mean by that? I can only think of two ways to think of it that way and do not know which is the most logical.
You give me any you want, as small as you want. No matter what, I can always find an in the natural numbers where, for every number bigger than it, the distance from each of the terms of my sequence from is less than your
Or, are you challenging me with an and asking me to show that for any number bigger than your , ( ), the difference between L and is less than all .
I know this question probably seems ridiculous but if I don't understand the easy stuff I won't understand the rest of the material.