I'm trying to do some revision for my analysis exam and when it comes to proofs where ε is used I dont understand what these proofs mean.
Like for example
Given
which is something to do with the limit of the sequence xn being l
why not use
?
analysis just makes no sense D:
In fact, all definitions related to limits contain these notations. The idea here is that for all , you can still find values of n larger than some N>0 (this should be added to your statement: Not for all n!)and where the difference between and the limit is less than that . In other words, the sequence is approaching the limit.
In fact, you can't just say because you may never find a value of that is equal to l. For example, if , then we claim that the limit of is zero as n goes to infinity. However, you can't give any number n such that . In fact, for any given , look at which can be made less than if you choose sufficiently large n, namely . This proves that the limit is zero! That is, for any given , you can find a natural N such that if , then
hope this helps!