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YellowSun

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Real Numbers - Yr 11 3 Unit Maths

Suppose a and b are positive irrational numbers, where a < b. Choose any positive integer n such that 1/n < b - a, and let p be the greatest integer such thar p/n <a.

Prove that the rational number p+1/n lies between a and b

Please help out.

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Real Numbers - Yr 11 3 Unit Maths

Year 11? Wow, that question is tougher than I'd have thought.

I'm assuming that means (p+1)/n, rather than p + (1/n). If it was the second case, I'm pretty sure the statement you're proving would be false.

Anyway, I'd do this by showing that (p+1)/n can't not be between a and b.

Let's see what happens if (p+1)/n < a. Then, p+1 is an integer such that (p+1)/n < a.
But p+1 > p, and so p is not the greatest integer such that p/n < a. So that case doesn't work.

(p+1)/n can be rewritten as p/n + 1/n, and substituting the two inequalities we got told into that, we can deduce that p/n + 1/n < a + (b-a) = b. Hence (p+1)/n < b.

Combining these two results, a < (p+1)/n < b.

In other words, you've just proved that there is always a rational number between any two irrationals! Go you!

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Why did the vector cross the road?
It wanted to be normal.