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For any positive real number x, prove ...

For any positive real number `x`, prove that there exists an irrational number `y` such that `0 lt y lt x`.

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RD SHARMA-REAL NUMBERS-All Questions
  1. Prove that 5-sqrt(3) is an irrational number.

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  2. Prove that 3+2sqrt(5) is irrational.

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  3. For any positive real number x, prove that there exists an irratio...

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  4. Show that the following numbers are irrational. 1/(sqrt(2)) (ii) 7s...

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  5. Show that the following numbers are irrational. 6+sqrt(2) (ii) 3-sq...

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  6. Prove that following numbers are irrationals: 4+sqrt(2) (ii) 5sqrt(...

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  7. Show that 2-sqrt(3) is an irrational number.

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  8. Show that 3+sqrt(2) is an irrational number.

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  9. Prove that ( 4-5 sqrt2) is an irrational number.

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  10. Show that 5-2sqrt(3) is an irrational number.

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  11. Prove that 2sqrt(3)-1 is an irrational number

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  12. Prove that 2-3sqrt(5) is an irrational number.

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  13. 4. Prove that sqrt(3)+sqrt(5) is irrational.

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  14. Prove that sqrt(3)+sqrt(4) is an irrational number.

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  15. If p ,\ q are prime positive integers, prove that sqrt(p)+sqrt(q) i...

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  16. Without actually performing the long division, state whether the fo...

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  17. Without actually performing the long division, state whether the fo...

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  18. Without actually performing the long division, state whether the fo...

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  19. Without actually performing the long division, state whether the fo...

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  20. Write down the decimal expansions of the following rational numbers...

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