Using the method of proof by contradiction, prove that if a product of two positive real numbers is greater than 1,000,000, then at least one of the numbers is greater than 1000.

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Omm2 Omm2 · Answer 1 · 2021-03-15T17:23:03+01:00

Step-by-step explanation:

To prove - Using the method of proof by contradiction, prove that if

a product of two positive real numbers is greater than

1,000,000, then at least one of the numbers is greater

than 1000.

Proof -

For a proof of contradiction , firstly we have to assume that the statement is not true.

Here given statement is - A product of two positive real numbers is greater than 1,000,000.

Let if possible, there does not exist 2 +ve integer real number such that at least one of the numbers is greater than 1000.

⇒both the number are in between 0 to 1000

Let a and b are 2 +ve real numbers.

⇒ab < 1000×1000

⇒ab < 1000000

As given,

ab > 1000000

Contradiction.

Hence proved.

If a product of two positive real numbers is greater than 1,000,000, then at least one of the numbers is greater than 1000.