Why we must get closer and closer to a value after any arbitrary point so as to consider that value its limit value? Why approaching a value means that we can get arbitrary close to that value?
Real Analysis
Given two sets:
$x = \{ 3, 2.5 , 2.04 , 2.03 , 2.02 , 2.001 , 2.0001, ... \}$
$y = \{ 4 , 6.25 , 5.76 , 4.25 , 4.025 , 4.001 , 4.00001 , ....\}$
As terms of the set $x$ gets closer and closer to $2$, terms of the set $y$ seems to be getting closer and closer to $4$. But terms of set $x$ also seems to be getting closer and closer to "values $\lt4$" like $3.999..$, $3.8999...$, etc. Then why do we say the $4$ is the value the terms of set $y$ seems to be getting closer and closer to rather then some value $\lt4$?
And also, why do we need to check for arbitrary number of terms to prove the existence of a value the terms of a set are getting closer and closer rather then just looking at a bounded (finite) number of terms of set? Why can't we prove the existence of a value the sequence is getting closer and closer just but checking finite number of terms of a sequence rather than looking at arbitrary number of terms of a sequence?
From the given set above, why is it that we say the value the terms of set $y$ is getting closer and closer to is $4$ rather than values $\lt4$ even though it does closer and closer to values $\lt4$?
Why, when formalizing the idea of a sequence getting closer closer to particular value, defined to check the existence of the value we are getting closer and closer to, do we need to look at arbitrary number of terms rather than finite number of terms?
So, my question is that why approaching a value means that we can get arbitrary close to that value, i.e, why getting closer and closer to a value means that we can get as close as we want to that value?
In book A Mathematical Bridge An Intuitive Journey in Higher Mathematics by Stephen Hewson

Aristotle: I see: all the terms in the sequence beyond a certain point
are closer to $1/2$ than any number I can specify. If I were to say that the
limit were anything other than exactly one half, then eventually all of the
terms beyond some point in your sequence would get closer to one half
than they would to my other supposed limit. I thus concede the point: the
sequence must tend to the limit of $1/2$.

Here he is saying about the fact the a sequence get closer and closer to many values until a certain point but diverges after that thus it can't be considered a limit value but it gets closer and closer to $1/2$ after any arbitrary point: that's why it is the limit value, can you explain to me why we must get closer to closer after any arbitrary point to consider it its limit value?
 A: 
Why approaching a value means that we can get arbitrary close to that value?

You show that the limit is closer to the sequence from some point on than any other real number - so if the limit of the sequence is some real number then it'd have to differ from the value you're comparing it with by some amount $\delta$ that's smaller than any other real number but still bigger than $0$. But in the real numbers there is no such $\delta$ - so they have to be equal.

why can't we prove the existence of a value the sequence is getting closer and closer just but checking finite number of terms of a sequence rather than looking at arbitrary number of terms of a sequence?

Because it can't get "closer and closer" with only a finite number of elements. Let's say you choose some finite set $A$ of elements and compare them to the limit $\ell$. Then the set $\{|a-\ell| : a \in A\}$ will also be finite and thus have a minimal value $m$. So you could never closer than $m$ to $\ell$. Are you familiar with the concept of the supremum and how it is fundamentally related to the real numbers?

But terms of set x also seems to be getting closer and closer to " values < 4 " like 3.999.. , 3.8999... etc. Then why we say the 4 is the value the terms of set y seems to be getting closer and closer to rather then some value< 4 ?

Because we can show that no value in this sequence will ever be smaller than $4$ - so we'd have to make a discontinuous "jump" to arrive at a value smaller than $4$ which would not be very natural.
Essentially you can modify any sequence at an arbitrary finite number of points and it'll still converge to the same value than before because there'll always be some last modification and after the point of this modification the modified sequence is exactly equal to the original one.

Here he is saying about the fact the a sequence get closer and closer to many values until certain point but diverges after that thus it can't be consisdered a limit value but it gets closer and closer to 1/2 after any arbitrry point that's why it is limit value ,can you explain me why we must get closer to closer after any arbitrary point to cosisder it its limit value?

That's just what it means to be a limit of a sequence. Stating the limiting behaviour is essentially a statement about the very long term behaviour of the sequence. Consider the sequence $n \mapsto \frac{1000}{n}$, surely you'd agree that this get's closer and closer to $100$ for a while, but it's not a "stable attractor" if you will, because we can also show that we'll start moving away from this point after a while, so maybe the limit is $1$ because it's moving towards that point even longer - but no, we run into the same problem of the distance eventually getting larger again. The only value that we never stop moving towards (in the grand scheme of things) is $0$ - and that's why we call $0$ the limit of this sequence - anything else would just be unnatural.
A: 
Given two sets:
$x = \{ 3, 2.5 , 2.04 , 2.03 , 2.02 , 2.001 , 2.0001, ... \}$
$y = \{ 4 , 6.25 , 5.76 , 4.25 , 4.025 , 4.001 , 4.00001 , ....\}$
As terms of the set $x$ gets closer and closer to $2$, terms of the set $y$ seems to be getting closer and closer to $4$. But terms of set $x$ also seems to be getting closer and closer to "values less than $4$" like $3.999...$, $3.8999...$, etc. Then why we say the $4$ is the value the terms of set $y$ seems to be getting closer and closer to rather then some value $\lt4$ ?

There is an issue here. You cannot simply talk about the limit of a set relative to the limit of another set. This idea does not work. What if I wrote $x=\{2.0001, 2.001, 2.02, 2.03, 2.04, 2.5, 3,...\}$ instead? The elements are exactly the same: I have not changed what $x$ is at all. All I did was list the elements in a different order of writing. Yet now, you would not say the elements of $x$ approach anything in particular, would you?
What you need here is a function $f$ from the set $X$ to the set $Y$. What you need in this case is $f(3)=4$, $f(2.5)=6.25$, $f(2.04)=5.76$, $f(2.03)=4.25$, $f(2.02)=4.025$, $f(2.001)=4.001$, $f(2.0001)=4.00001$, etc. Now, the idea you want to talk about makes sense. Having taken care of that, your question can now be answered partially.
Here is the issue: we cannot say that the limit is 4 here. You are correct in being skeptical here. Do you know why we cannot say that? Because the sets you have provided us in your example, $X$ and $Y$, have only finitely many of their elements specified. With only many of the element specified, it means that the infinitely many other elements in the set are unknown: since they were not specified, we do not know what they are. The limit of $f$ could actually be anything, or it could not exist at all as well.

And also, why do we need to check for arbitrary number of terms to prove the existence of a value the terms of a set are getting closer and closer to rather than just looking at bounded (finite) number of terms of the set? Why can't we prove the existence of a value the sequence is getting closer and closer just but checking finite number of terms of a sequence rather than looking at arbitrary number of terms of a sequence?

A sequence cannot be uniquely defined by only specifying finitely many terms. Here is an example: look at the terms $1,2,4,8,16,...$. Can you find the next term in the sequence? Say it with me. The next term is... $31$! Wait, what? Not $32$? Correct. I assume you were thinking that the pattern I was specifying is the pattern of the powers of $2$. But there are infinitely many patterns and sequences who first five terms are $1,2,4,8,16,...$. Saying what the first five terms are does not tell me what sequence you are working with. The only way to uniquely specify a sequence is to give a formula for that sequence, implicit or explicit. And as such, since the sequence is not uniquely specified by only finitely many terms, the limit is not uniquely specified either.
And this is why the definition of a limit requires knowledge of infinitely many of the terms in the sequence. With finitely many terms, it may look like the sequence is approaching something. But it could be that all of the terms that comes after the finitely many you looked at begin to behave chaotically, and so the sequence does not approach anything. So for your example, you had that $f(2.0001)=4.00001$. But it could be that $f(2.00001)=67.4444$. This is possible, and yet, you could not have known that this happened, because you chose to only look at finitely many terms, rather than at all of them. Now, with this new information, you would change your mind about $f$ approaching $4$. With only finitely many terms, it is unclear whether $f$ approaches anything at all.

So, my question is that why approaching a value means that we can get arbitrary close to that value, i.e, why getting closer and closer to a value means that we can get as close as we want to that value?

You are essentially asking "why is this the way it is?" If you are getting closer and closer to a value, then, of course you are getting closer and closer. This is a tautology. A better question to ask is, "why do we require that a number be approached arbitrarily to call it a limit?" The answer to this is because this is what mathematicians found was useful. Here is the thing: if you cannot get arbitrarily close to a value, i.e, if there is a minimum distance between the variable and that value that cannot be breached,  then intuitively and geometrically, in what sense are you even approaching that value? Answer: none, because if there is some small distance that cannot be breached by the variable to get closer to the value, then that variable and that value are like parallel lines: they will never get closer. So then it is not useful to say that the variable is approaching the value in that case. This is why we require that it can get arbitrarily close. That is what captures the core intuition behind limits.
