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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???

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    There is no such number – 2017-02-09
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    then there must be a gap on the real line – 2017-02-09
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    There is no gap. It is the fact that there is no gap which implies that there is no unique "next" number. If there were a "next" number, there would be empty space between $x$ and the next number after $x.$ – 2017-02-09
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    cant understand your argument – 2017-02-09

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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 $athat 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:

and

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.

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    There is no real number immidiately next to a real number? Perhaps there is some number we cannot discover yet – 2017-02-10
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    @MuhammadTahir It's not that we haven't discovered all the reals - we can *prove* that between any two reals, there is a third. Namely, if $a${a+b}\over 2$ is between $a$ and $b$. So it's not that we haven't *found* the "next real", it's that it *doesn't exist*. Do you understand why this is true? – 2017-02-10
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There is always a real number between an two distinct real numbers $a$ and $b$. For example $\frac{a+b}{2}$.

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    so whats the conclusion? – 2017-02-09
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    there is no such thing as an immediate successor. – 2017-02-09
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    then what is 'next' to one? a number or a hole? – 2017-02-09
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    neither, it does not exist, there is no such thing as the next one. The same thing happens with water after it stops becoming drops. – 2017-02-09
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    nor a number nor a hole then whats the possibility? – 2017-02-09
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    @MuhammadTahir Another possibility is that the concept "next to" does not make sense for real numbers. – 2017-02-09
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    set or real numbers is linearly ordered by less than – 2017-02-09
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    @MuhammadTahir But this does not give us the concept of "а real number next to а real number". How would you define a number next to a number $x$? – 2017-02-09
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    I cannot define this thats why i asked for any idea – 2017-02-09
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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.

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    then there is hole? – 2017-02-09