1. The problem statement, all variables and given/known data *Sorry I could not get the math symbols to work properly so I did it by hand. I hope this isn't too much trouble. Prove: | sqrt( x ) - sqrt( y ) | <= | sqrt ( x - y ) | for x, y >= 0 Hint: Treat the cases x >= y and x <= y separately. I am new to proofs and we can't use calculus. It's all confusing to me and we've only just begun. The farthest we got was to the Triangle Inequality / the AGM Inequality so I assume that's the most we can do. 2. Relevant equations I assume whomever is helping me already knows the AGM and Triangle Inequalities since they're the most basic of all the inequalities for proofs. 3. The attempt at a solution I squared both sides and moved the absolute value signs to the individual variables and then in then in either case, I can remove them. On the left side I had x + y and just removed that and -2 sqrt(x) sqrt(y) and removed that one too. On the right I had x - y so I write it as that if x > y and -y + x if x < y Subtract x from both sides and then y - 2 * sqrt( x ) * sqrt( y ) <= -y I am pretty lost here. Iff x >= y then the roots would be the same and I could replace sqrt( y ) with sqrt ( x ). I would then add y to both sides and thus 2y - 2x <= 0 2(y-x) <= 0 and that's not true since x >= y and the equality would only hold true in that situation if they were equal, which we know the equality is true if y > x from substitution PS. Thank you for taking your time to review my question and for whatever help you provide. I appreciate it very much, this question frustrated me for far too long so far.