Please help me understand definition of convergent sequences?

It's not zero. The definition of a limit is similar to that of a convergent series. Actually here it is:

We say that lim[x->∞] f(x) = L, where L is a real number and f is a function defined for all reals if

for all ε > 0, there exists M such that |f(x) - L| < ε for all x > M.

See, it's exactly the same, except that x and M are real instead of integers.

The reason for the epsilon thing is that the value may not be exactly the same. Look at what

|a(n) - L| = 0

means. It means a(n) = L. So you're saying the sequence is just constant past some point n = m. Well, that doesn't make any sense because we know the sequence 1/n converges to 0, but it can never actually equal zero, no matter how large we make n. So instead we say that

|a(n) - L| < ε, for all n > m

or that "the distance between the values in the sequence 'a' for every n past some value m is less than an arbitrary non-zero number."

Geometrically what this means is that you're trapping the "tail" of the sequence between two values, L - ε and L + ε. So we're basically ignoring the first m terms, a finite amount, and saying that the rest of them, all infinitely many, fall into this window. So instead of considering the sequence itself, we can consider that value. Similar to how, in physics, instead of considering an entire object's mass you just consider it's mass as being concentrated at it's center. It makes the problem easier, but doesn't change the fundamentals of it.

What making ε arbitrary does is ensures the window formed by L - ε and L + ε can be as small as possible without actually condensing it to something we can't "look through." But the key is that it can be very, very, very small. Smaller than you can imagine. So small that not even a quark (the particles that make up atoms) could fit through it. So realistically we could make it so small that we couldn't look through (since a photon, a particle of light, is larger than a quark), but mathematically there's still a window. This gives the possibility I mentioned before. The sequence doesn't actually have to attain it's limiting value, as is the case with 1/n. It just gets "arbitrarily close." That's basically just a fancy way of saying, "yea, they're not equal, but they might as well be." Which is why we can treat the end of the sequence 1/n as just being 0.

Now, in practice, how much you can treat as 0 depends on ε, which in the real world basically becomes your error. So you have to decide how much you want to be off by. In mathematics, we don't want to be off by any, which is why we do this for ALL ε > 0. We mathematicians can achieve an error much smaller than any instrument that could ever be built in the real world.

So you've done sequences in pre-calc, right?

And you've done limits in calculus.

So think of it this way:

A sequence is really a function except you only evaluate it at integers (Not iterating functions).

So if the sequence converges, it means that the limit exists when you "use" the sequence in infinite amount of times.

Now what you have is the epsilon-delta definition of the limit.

So what it's saying in Layman terms is that no matter how big x gets, the sequence approaches a specified amount.

Now |a(n) - L| never actually reaches 0. It just approaches it.

Also, if I remember correctly, the "m" is supposed to represent the number that n has to be larger than in order to converge.

ϵ is assumed as a constant.

Why can't it be simply |a(n) - L| = 0?

because even for a converging sequence, where the consecutive term is less than its predecessor, it will never reach to a point where one of its term is 0. thus, ϵ is considered to be a very small number which is almost equal to zero but will never be 0.

for example:

5, (0.1)5, (0.1)^25,..........

it is a converging sequence. the 10th term of this sequence will be,

a10 = a*(r^n-1) = 5 * ( 10 ^ -10 ).

this is a very small value, but certainly not zero.

I didn't understand the use of "m".

Umm, Harald, you have misstated the Cauchy criterion. The criterion demands that for each t > 0 there exists an integer N such that | u_m - u_n | < t for m, n > N. interior the specific occasion u_k = a million + a million/2 + ... + a million/2^ok. for this reason, if m > n u_m - u_n = a million/2^(n+a million) + ... + a million/2^m. So, permit t > 0, and decide N using fact the smallest constructive integer extra desirable than -log(t)/log(2). Then for m, n > N, |u_m - u_n| =a million/2^(n+a million) + ... + a million/2^m <= a million/2^(n+a million) + ... + a million/2^m + ... = a million/2^n < a million/2^N < t