Hello, I am new to doing mathematics of the type in Apostol. I wish to study analysis. Hence I have started with Apostol Calculus volume 1 so as to build my fundamentals. I need a tip in solving the exercise problems which one encounters in first few chapters. I am presently on the introductory chapter. My query is, while proving a theorem such as :- 'there exists a real number z such that x < z < y for any arbitrary x , y' , is it ok to prove that z = (x + y)/2 meets the condition. Apostol never really makes use of '2' in his proofs upto this point. For this I will have to first prove that 2 > 1 , I think. Or should I stick with arbitrary real numbers such as 'a' . This issue is also relevant while proving that there is an irrational between any two rationals. Can I make use of √2 in such cases ? I will prove √2 is irrational and then proceed to make use of it and its reciprocal. For a beginner level such as mine , are such proofs considered valid ? And at what level should one strive for elegance ? I had browsed some analysis books in the past and I found they never make use of particular numbers such as '2' and '√2' and rather make use of general properties of the numbers involved.