Please indicate the method of proof and begin each sentence of the proof on a new line.Result 2If x and y are positive real numbers, then Jx+y # Vx + Jy .Result 3There are no positive integers m and n such that m +m +1=n² .Result 4Prove that 3" > n° for all positive integers n.Result 5For every real number x +1,1-x"n-11+x+x* +x° +...+x"for all n eN.1-x

Hey, since there are multiple questions posted, we will answer the first question. If you want any specific question to be answered, then please submit that question only or specify the question number in your messageResult 2 If x and y are positive real numbers, then x+y≠x+y. We use the method of contradiction to prove the above result. If possible suppose that x+y=x+y where x and y are real numbers. Then, x+y=x+y⇒x+y2=x+y2⇒x+y=x2+2xy+y2⇒x+y=x+2xy+y The last equation has an extra term 2xy in the right hand side than the left hand side. So, x+y=x+2xy+y does not hold if both x and y are non zero real numbers. Thus, x+y=x+2xy+y does not hold if x and y are positive real numbers and we reach a contradiction. Hence, our assumption that x+y=x+y is true is wrong. Therefore, x+y≠x+y. So, If x and y are positive real numbers, then x+y≠x+y.