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prove the associative law of addition for real numbers.
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let a, b and c be real numbers.

(a + b) + c = a + (b + c) 

we will prove this using mathematical induction.

base case:

when n = 2, (a + b) + c = a + (b + c)

let (a + b) = x ⇒ a + b + c = x + c ⇒ (a + b) + c = a + (b + c) 

inductive step:

assume that (a + b) + c = a + (b + c) for n = k, where k ≥ 2.

let (a + b) = x ⇒ a + b + c = x + c 

let (x + c) = y ⇒ (a + b) + c = y 

similarly, a + (b + c) = y  ⇒ (a + b) + c = a + (b + c)

therefore, for all real numbers a, b and c, (a + b) + c = a + (b + c).
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