It is very easy to prove it, using your definition. Consider an open interval of length a > 0. Consider a natural number N such that 1 / N < a. Then your interval contains already two rational points, of the form k/(2N) and (k+1)/(2N). But between two rational numbers, there are already an infinity of rational numbers (take the middle, the middles of the halfs, etc...), so your interval contains already an infinity of rational numbers. Finally, every neighborhood of a real number contains an infinity of rational numbers. So every real number is an accumulation point of Q.

Dear Kulthoum, to verification that Q is dense in R, we can see the proof in "Mathematical Analysis  or Introduction to Real Analysis"  text books.

See for example 

Rudin's "Principles of Mathematical Analysis".

Not only to rationals but to irrationals as well - i.e.,  every real number is equally surrounded up close by a neighborhood, how small it might be, that has infinitely many irrational numbers as well. 

To offer yet another explanation that coheres with the above explanations yet perhaps offers something distinct not by answering the question per se but by noting something about the reals vs. the rationals that perhaps sheds light here:

It is well known that the rationals are infinitely dense. Between any two rational numbers, there exists infinitely many rational numbers. Yet we all know the rationals are countably infinite. A major problem in mathematical analysis and the development of calculus was realized when mathematicians began to see how the informal construction of the reals could not serve as a basis for analysis. A rigorous definition of limits and a rigorous construction of the reals was required.

A major development in mathematical analysis came once mathematicians were able to construct the real number line first in terms of limits of or boundaries of rational numbers. That is, they were able to formally define the entire set of real numbers rather than simply noting that certain numbers such as pi were irrational. Also, Cantor's proof was instrumental not only in establishing the diagonal proofs used by many other mathematicians or his foundation of set theory, but in particular his proof that not all infinite sets were equal.

Only when a rigorous formulation of the reals existed was such a thing as a "ball" sufficiently well-defined to enable the vast majority of modern mathematical proofs in analysis, topology, etc.

The number line can only exist via the limits of rational nu8mbers. In between any two rational numbers there exists a set of infinitely many irrational numbers greater than the entire set of rational numbers. Without this understanding, fundamental notions such as continuity couldn't exist.

"Without this understanding, fundamental notions such as continuity couldn't exist"

Continuity is equally meaningful for functions defined on the rational numbers.

The basic difference between the rational line and the real line is that the latter is connected. In 1925,  P.Urysohn proved that a countable regular space cannot be connected (though R.H.Bing gave an example of a countable connected Hausdorff space in 1953). So, under weak assumptions, connected topological spaces must be uncountable. This is perhaps the best explanation for the massive growth of additional numbers when passing from the rationals to the reals.

If you look at any standard textbook on calculus, you will notice the outstanding importance of connectedness through Bolzano's Intermediate Value Theorem for continuous real functions and through Taylor/Lagrange's Mean Value Theorem for differentiable real functions.

While the integers and the rationals have the same cardinality (i.e. are bijective), they have a different metric. It is only in the "usual" metric d(x,y) = |x - y| that the rationals embed in the reals in a metric preserving way  and are dense. In fact the reals can be defined as the equivalence classes of Cauchy sequences in Q,  where two sequences are equivalent if their difference sequence converges to 0 (i.e. for all epsilon > 0 the difference sequence is eventually smaller than epsilon ). Essentially this means that the rationals are dense in the reals by definition, because you constructed them that way. You force any Cauchy sequence in Q to converge by pulling out the limit out of the hat of set theory.    

You need not, however,  put the usual metric on the rationals. Other good metrics are the p-adic metrics defined by

|x|_p = |a/b|_p = |p^k c/d|_p = p^{-k}

where a,b,c,d are integers and c,d are relatively prime to p.  This is important in number theory, because the p-adic metric essentially says that two numbers are close if the difference is divisible by a high power of p.This is a perfectly good metric that satisfies the usual axioms ( e.g. |x + y|_p \le max(|x|_p, |y|_p) \le |x|_p + |y|_p). The Cauchy sequences in THIS metric certainly do not converge in the reals. However just as you can construct the reals as equivalence classes of Cauchy sequences, you can construct the p-adic numbers Q_p  as equivalence classes of  Cauchy sequences with respect to the p-adic metric. They are in many respects similar to the reals (locally compact), and you can do standard analysis things like integration, but they are somewhat weird in other respects (e.g. not being locally connected). They are a big thing in certain branches of number theory, representation theory and algebraic geometry. For example the p-adic numbers and more particularly Q_3 is used in an essential way in the proof of the Fermat theorem by Andrew Wiles. 

It seems that you are worried about the cardinalities. But there is no problem: The rationals are countable, right, but from a countable set you can select uncountably many sequences. So there are enough sequences to converge to every real number.

Thank you all for the useful answers!! . But yeah Dr. Rainer this is what i am worried about "the cardinalities". We know that the rationals are countable set as well as naturals so why do the rationals have R as an accumulation point while naturals don't . . Thank you for your answer.

Let x be arbitrary real number and every open interval (open set) containing x must contains infinitely many rationals and hence every real number is a accumulation point  in real space. Thus d(Q)=R

Ismail, I would like to add that you are in excellent company with your worries. When Georg Cantor first discovered the possibility that there are distinct infinite cardinalities, he was in the habit of exchanging his ideas regularly with Richard Dedekind. From their mutual letters we know that both men were extremely puzzled and bewildered by the strange phenomena of infinite sets such as space-filling curves etc.

Thank you so much .

if you get any irrational number q  there exists a sequence of rational numbers converging to q. This implies that any irrational number is an accumulation point for rational numbers. For any rational r consider the sequence r-1/n. Hence r is an accumulation point of rarional numbers.