what is the smallest unit of a real number to which it is composed of? I searched for relevant questions but my point is different. For example take set of real numbers with usual order then is the immediate successor of one. Firstly, I ask that is there any such number?? If yes then how surprisingly we beleive existence of a number but we cannot see that???
 A: As the other answers have shown, the real numbers are dense - between any two, there is a third, so that there is never a "next real number".
You write in response to this:

then there must be a gap on the real line

and I think this is really the "meat" of the question.
Actually, the situation here is the opposite - if there were real numbers $a<b$ with no real in between them, that would be a hole! There'd be a "gap" between $a$ and $b$, where there should be a real but there isn't.
In fact, in a precise sense, the reals have no "holes" - see this article for example, or better yet Dedekind's essay on constructing the reals from the rationals by "filling in" the holes. There are multiple ways to describe the "fullness" of the reals; the two most common ones are:


*

*the least upper bound property: any nonempty set of reals with an upper bound has a least upper bound


and 


*

*the completeness property: every Cauchy sequence converges.



I think what's confusing you is the following: we can define what "next real number" means, and yet they don't exist! So doesn't this represent a "gap" in the reals, in the sense that some possible behavior doesn't occur?
Note: If that's not what you mean by "gap", then what do you mean?
Well, the problem is this: just because you can describe something, doesn't mean it should exist. For instance, I can also imagine real numbers for which multiplication isn't commutative: that is, reals $r, s$ with $r\cdot s\not=s\cdot r$. Yet multiplication of reals is commutative! There are no such reals $r$ and $s$. And in fact, this is a property we want $\mathbb{R}$ to have, similarly to how the field axioms - which imply that there are no successive real numbers - are properties we want $\mathbb{R}$ to have.
If you don't find this satisfying - and I suspect you won't at first - try making precise what you mean by "gap." I think in your efforts to do so, you'll see what I'm getting at.
A: There is always a real number between an two distinct real numbers $a$ and $b$. For example $\frac{a+b}{2}$.
A: There is not such a number. Arguing by way of contradiction, for given $r\in\mathbb{R}$, suppose that $r_0$ is immediate successor of $r$. Then, $r<\frac{r+r_0}{2}< r_0$; a contradiction.
