Proving a limit of at infinity based on certain given stipulations?

Heres the question that I think I have figured out. I was hoping somebody could look over my proof and tell me weather it is valid. Anyway, the question is:

Quote:

__LIMIT PROBLEM__
If

for all

,

and

then prove that:

Okay, so heres my attempt at a proof. Please point out any errors you may see or find:

Quote:

__MY ATTEMPT AT PROVING THE LIMIT__
By the definition of a limit, we know that:

Means that for every:

,theres is a

such that:

implies:

The above inequality implies that:

Also, since:

Then it follows that (by multiplying

to both sides of the g(x) inequality):

[I don't know if the follwing is neccesary for this proof, but better safe then sorry]

(We need not worry about

being negitive; since

for all

, then the denominator is always positive. And since, by the stipulations made by the problem,

for all

obviously means that

for all

; then it folllows that both the denominator and numerator are positive, therefore

for all

)

To prove that the limit approaches infinity, we must show that:

Implies that, for any

we have:

.

But we are already there! (atleast I think, as long as there are no errors) From the given limit of g(x), we were able to show that :

So letting:

and

and rememebering the stipulations made by the problem of the value of f(x), then the train of implications goes like this:

Remeber that:

So:

Which shows that:

For:

and

FINNALY, this all means that:

(granted that the stipulations on f(x) are given)

Now, are there any errors in this proof? And is it always neccesary to mathematically go over every detail in a proof? Like, is my proof to long? Is it okay to make simple assumptions that your thinking isn't neccesary? (Like stating or not stating in the proof that for all ) Thanks in advance