1. The problem statement, all variables and given/known data Prove that there is an additive identity 0∈R^n: For all v∈R^n, v+0=v 2. Relevant equations Axiom of Real Numbers: There is an additive identity 0∈R : For all a∈R, a+0=a and o+a=a 3. The attempt at a solution Solution 1 (My own attempt) : Let v=(v1, v2, v3... vn). Then by axiom of R (as stated above), for every vi (i=1, 2...., n) there exists 0∈R such that vi+0=vi. Thus, there exists 0=(0,0,......0)∈R^n. Solution 2 (Professor's Attempt) : Let 0=(0,0,....0)∈R^n then v+0=(v1, v2 .... vn) + (0,0....0) =(v1+0, v2+0, .... vn+0) =(v1, v2 .... vn) - by axiom of R (as stated above) =v However, I think solution 2 has a problem that it assumed existence of 0 when the question is asking to prove its existence. Or is my thought wrong? And also, is there anything wrong with solution 1?