Heres the given problem:

and:

then:

I am going to put my proof below. I'm going to ask the reader for a big favor; if you happen to see anything at all (and I realy do mean anything at all; even if you just see something that is only the slightest bit incorrect, please still point it out) if you see any mistakes, please nit-pick and split-hairs, and say "well technically thats not...", ect., I think you get the picture. I'd like to be able to one day complete proofs that have the qulaity of near perfect rigor; and I can't do that without some incredibly precise and thorough criticisim (Wait). Heres my attempt at a proof:

Thank you in advance for any errors or 'technicalities' you may find, and for any advice you can give. ThanksQuote:

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.... (Worried)) 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.