For any positive integer x prove that there exists an irrational no. y

1.	For any positive integer x prove that there exists an irrational no. y such that 0
Answer» If x is irrational, then {tex}y = \\frac { x } { 2 }{/tex} is also an irrational number such that 0 < y < x.If x is rational, then\xa0{tex}\\frac { x } { \\sqrt { 2 } }{/tex}\xa0is an irrational number such that\xa0{tex}\\frac { x } { \\sqrt { 2 } } < x{/tex}\xa0as\xa0{tex}\\sqrt { 2 } > 1{/tex}.{tex}\\therefore \\quad y = \\frac { x } { \\sqrt { 2 } }{/tex}\xa0is an irrational number such that 0 < y < x.Hence, for any positive real number x, there exists an irrational number y such that 0 < y < x.

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