I think you meant |x(n) - L| < epsilon in your last inequality.
What do we mean when we say that a sequence of real numbers converges to L? Well, what do we want it to mean? At first, we may think that what we mean is that, if we start listing the numbers in the sequence: x(1), x(2), x(3), ..., that we will eventually find x(infinity) = L. But x(infinity) does not exist in a concrete sense, because we can never list out an infinity of numbers. Thinking a little more, we realize that what we really want to say is that, as n gets larger and larger (i.e. as n tends to infinity), we want x(n) to get closer and closer to L. That fits our intuition that in the limit, x(infinity) = L.
What do we mean by "closer and closer?" Does that mean, for example, that each successive term in the sequence must be closer to L than the previous one? That is, do we want it to be necessary that |x(n+1) - L| < |x(n) - L|? It turns out that this is too restrictive. What about the sequence 3, 4, 100000, 3, 3, 3, ...? Our intuition tells us clearly that this sequence converges to 3, but 100000-4 > 4-3, which wouldn't satisfy that definition. You see, what only matters is what the sequence does as n gets closer and closer to infinity; because it's never possible to reach infinity, there is a lot of wiggle room. The sequence can behave erratically for the first million, the first billion, even the first googol terms so long as it "settles down" as n gets even larger.
But now we see why, in the definition of the limit, there is the phrase "there exists some a natural number, Kappa, such that for all n greater than or equal to Kappa, ...". All we need is for the sequence to "settle down" for sufficiently large n. We don't know how large Kappa should be--it could be one million, one billion, even one googol. But it clearly must exist; otherwise erratic jumps can occur forever, and our intuition tells us that the sequence does not get "closer and closer" to L in any fashion.
So how do we put into symbols the idea of "get closer and closer" and "settle down" that I brought up before? We return to the idea that each successive term must get closer to L and tweak it a little by focusing on the question of "how close?" Intuitively, when n = infinity, we want x(infinity) = L, so that |x(infinity) - L| = 0. When n is not infinity but as it gets closer to infinity (that is, as n -> infinity), we want |x(n) - L| to get smaller and smaller, so that x(n) becomes closer and closer to L. Indeed, for each level of closeness, let's call it epsilon, we want |x(n) - L| < epsilon for n big enough. How big? Well, Kappa(epsilon). I write it like this to show that Kappa is a function of epsilon (the level of closeness). Beyond Kappa(epsilon), the sequence can still behave erratically as long as it remains within epsilon of L. By making epsilon the independent variable, we make rigorous our idea of "settle down," and we can allow for arbitrarily long lengths of erratic behavior as long as there is some cut-off point.
Finally, we want convergence to mean that the difference between x(n) and L eventually become closer and closer to 0, so that in the limit, the difference is 0. This is what we do when we say that for all epsilon > 0, we can find a cut-off point Kappa, such that if n > Kappa, the sequence remains within epsilon distance from L. We can choose epsilon, no matter, how small, and find a Kappa (usually larger as epsilon gets smaller) such that the sequence is bounded by |x(n) - L| < epsilon when n > Kappa.
Hence, "A sequence, X, in the real numbers is said to converge to some L, or L is said to be the limit of X, if for every epsilon > 0 there exists some a natural number, Kappa, such that for all n greater than or equal to Kappa, the terms, x(n), of X satisfy |x(x) - L| < epsilon."
