First, lets make things crystal clear. This is what must be shown to complete the proof:
[1-a]
[1-b]
Now, next thing I shall do is to list a few properties that I will be using in my proof, so I need not explain them in the middle of a logical flow, but rather I can referance to them:
[i]
- For any numbers
and
, the follwoing is true:
[ii]
- For any numbers
; if we know that:
then the following inequality holds:
[iii]
- For some mathematical statement
, and some different mathematical statment (which is a function)
, and arbitrary numbers
and
(and taking note that the expression denoted by
is
) ; with all the previous stipulations assumed; and, lastly, knowing the two following statements to be true:
and that:
Then it follows that (I think it follows from the above....
) it follows that:
Now, with those 'properties' out of the way, let me begin the proof. From the given information and the definition of a limit, we know that for every
there is some
such that:
[2]
The problem also gave us the fact that, for the denoted
,
[3]
So, combining [2], [3], [i], and [ii] we can conclude that:
[4-a]
[4-b]
Now, as we can see, the above implication between the two inequalities looks allot like what we are trying to prove, [1] that is. Now, we know from [iii] that we can get to [1-a] from [4-a] by simply letting:
And, we can also easily get from [4-b] to [1-b] by letting:
So, from all of the above, we have been able to show from the given information that:
[1-a]
[1-b]
Which, by the definition of a limit, means that:
I think, if I've made no errors above (but, ofcourse, I could have very easily made many errors above, I'm no expert), but if this is correct, I think that completes the proof.