Htykuut

I know what a convergent sequence is and how it works and everything but whenever I look at the definition my mind cant suddenly make the link with my intuition part of the brain so I realise that I am actually going over the same thing in my brain and still get no good answer for my questions about the definition. A sequence $x_n$ of real numbers converge to a point $L$ if given $epsilon>0$, $exists$ $n_0=n_0(epsilon)$ such that $$ |x_n-L| < epsilon quad for everyquad n > n_0. $$
My first question is why do we use this definition? My second question is what does each term exactly mean and in general how did you come to digest this thing in your brain. I do seem to have general idea of it but i want to be precise and would like some directions and explanations. THANKS.

M: Here is a good sequence $(x_n)_{ngeq1}$ for you. It converges to $pi$. For $10$ dollars it's yours.

E: I looked at your $x_5$. It's $=4$. That's far away from $pi$.

M: What precision do you have in mind?

E: The error should be at most $0.01$, say.

M: Look here at this $x_{53}$ for example. It's $=3.14367$.

E: Maybe I should lower my tolerance to $0.0001$.

M: No problem. Let me check. Here it is: $x_{1538}=3.141512$.

E: I'm not convinced. These could be numerical coincidences.

M: There is a warranty booklet coming with the sequence. It contains a proof that the error is less than $0.0001$ for all $n>1600$.

E: How about even higher precision?

M: For only five more dollars you get the guarantee extended to $0.000001$. If I remember correctly the corresponding paperback contains a proof that the error is less than $10^{-6}$ for all $n>600,000$.

And on, and on.

I think about the $epsilon$ as sort of an "allowable error" term. You specify how much error you are willing to allow: I want all the points to be within $epsilon$ of $L$. Then we say the limit of the sequence is $L$ if you can find a point after which all the terms are "acceptable" in the sense that they are not too far away. The important thing is that you have to be able to find the point at which they are all acceptable regardless of what the "allowable error" is, as long as you allow some wiggle room (i.e. $epsilon$ cannot be $0$).

As to why, there must be a precise definition since otherwise it would not be possible to prove much about convergent sequences---they would fail to be a useful mathematical tool.

As for the meaning, you can visualize this somewhat simply. Plot a point on a number line and then start plotting a sequence of points that converges to the original point.

Now put an interval centered at the original point. The definition says that from some term onward, every point you plot lies in the interval.

That is, from some term ($n_0$) onward ($n ge n_0$), every point ($x_n$) lies in the interval ($|x_n - L| < epsilon$).

The OP asks

My second question is what does each term exactly mean and in general
how did you come to digest this thing in your brain.

Each term on its own means nothing. The sequence $(x_n)_{,n ge 0}$ is 'carving out a story' and the value of any term $x_k$, for a fixed $k$, says nothing. It is how the sequence 'continues, and keeps acting' at each subsequent term that imparts the 'message'.

If $w$ is any real number and the sequence $(x_n)_{,n ge 0}$ 'never stops showing up' with values $x_m$ that can get arbitrarily close $w$, then $w$ is said to be a cluster point. If the sequence is bounded it must have one cluster point, and if there is only one 'it keep approaching', that number is called the limit point.

I offered the above intuitive verbiage to describe how one might be able to 'digest this thing' into the brain. But to do it formally, you need to be able to understand logical quantifiers and be comfortable with real numbers.

Of course the reason investigating sequences is so rewarding is that the real line has no gaps (it is a 'complete' system of numbers).

The notion of a limit a sequence $(x_n)$, in simple terms, is a point $L$ such that all of the later terms $x_n$ in the sequence are arbitrarily close to $L$. Geometrically, this means that, given any small $varepsilon>0$, we can find a large integer $N$ such that, whenever $ngeq N$, we have $x_nin[L-varepsilon,L+varepsilon]$.