Question

Given 2 real number a,b.
Prove that if a 6 years ago

Answer 1

I have the solution for picking a rational number r, such that a < r < b
But I got lost half-way through the proof:

Answer 2

Okay the general idea is to prove that m/n is between a and b.
So it replaced the rational number with integer/integer by the definition of rational number.

But how do I do it for an irrational number?
I know that rational + irrational = irrational

So I can probably add the smallest rational number between m/n and b
Such that:
a < m/n < t < b

Answer 3

\[a < \frac{m}{n} < t < b\]
where
\[ t = \frac{m}{n} + irrational \]

Answer 4

What if I did this:

and followed the steps in the proof...
Since I know rational + irrational = irrational.

Answer 5

Wait I'm an idiot,

I already proved that there exists a rational number between a and b such that a < r < b.
Then:
\[a - \sqrt{2} < r - \sqrt{2} < b - \sqrt{2}\]
and since subtraction is closed under real numbers
and since r - sqrt{2} is irrational
I have
real < irrational < real
LOL easy

Answer 6

Then again I have to write all the steps down in case I get this on the test