how does one prove the associative rule for addition of positive real numbers in elementary terms I want to know how do I prove the associative rule for addition in elementary terms. I searched about the proof on Google but I was not able to figure head or tail about it. So how does one prove the associative rule? 
 A: Associativity can't be "proven", only demonstrated. It is an assumption, one of the requirements for addition to be a group operation.
A: In case we have axiomatized:


*

*$\sum_{i=1}^n x_i=\sum_{i^1}^n x_{\sigma(i)}$ for a permutation $\sigma\in S_n$ (addition is commutative)

*$x+y+z=(x+y)+z$ (you evaluate addition from left to right - reading direction)


Then we can prove:
$$
(x+y)+z=x+y+z=y+z+x=(y+z)+x=x+(y+z)
$$
But it has to be based on axioms, after all.
A: In Rudin's book, Principles of Mathematics, the real numbers are constructed using Dedekind cuts with a '$\mathbb Q$' build. He defines a binary operation $=$ on the set of cuts, and does indeed prove that this operation is associative (it follows immediately from the fact that addition is associative on the rational numbers).
But, for the sake of argument, suppose you have constructed the real numbers by another means and that you have defined a commutative binary operation $+$ that is associative on a 'dense' subset $Q$. Let $x, y, z \in \mathbb R$ and suppose you can find $a,b,c,e,f,g \in Q$ such that
$\tag 1 a + (b + c) \lt x + (y+z) \lt (a + e) + [(c + e) + (f + g)]$
But then 
$\tag 2 (a + b) + c \lt x + (y+z) \lt [(a + b) + c] + e + f + g$
If you can take $e,f,g$ to be arbitrarily small numbers, then can 'squeeze down' so that  you must conclude that
$\tag 3  x + (y+z) = (x+y) +z$
