Hi NockWodz
I can say that we use epsilon delta proofs to prove that a limit exists because that's literally the definition of a limit.
Hence, to prove that some objects exists or is equal to some other well defined object, the way is to prove that it matches the definition of that object.
In mathematics:
Axioms, definitions, theorems are three important classifications of the formal structure, yes?
Here is the definition of an even number:
$$ 2 k \,\,\,\,\,\ k \in \mathbb{Z} $$
To prove that 50 is an even number, you have to prove that:
$$ 50 = 2 k \,\,\,\,\,\, k = 25 \in \mathbb{Z} $$Hence it is even. Likewise the limit of a function is another object like the even numbers, and you have to prove that a function has a limit, ie, prove the definition fits.
In most real analysis texts, the definitions are developed from sequences and series then moved onto continuous and discontinous real valued functions.
So now, the definition for the limit of a sequence is roughly " A sequence a sub n is convergent to X if the limit as n goes to infinity the absolute value of the difference between a sub n and X is less than all epsilon where epsilon is greater than or equal to zero."
IIRC:
$$
\forall \epsilon \gt 0 \,\,\,\,\, \exists M \,\,\,\, \text{S.T} \,\,\,\, \forall_{n} \ge M \,\,\,\, |a_{n} - X | \lt \epsilon
$$
Thats what I recall for now being the definition of a limit for a convergent sequence, anyway to prove that any sequence converges to some limit, you have to prove that the sequence matches the definition given.
So its like that.
Of course, there will be many equivalent definitions of one mathematical object, so you can choose the most complicated definition or the simplest, and I think the epsilon delta definition is probably the simplest possible, there is some topology here which basically has a higher level definition of a limit, I think, but you can see what I am getting at.