I asked a student to show that between any two rationals is a rational.
She did the following: if x < y are rational then take δ << y-x and rational and use x+δ
This is correct though more complicated then what I had in mind: (x+y)/2
I then asked her to prove that between two irrationals is an irrational.
She did the following: if x < y are irrational then take δ << y-x and rational and use x+δ
1) Which proof for rationals is better:
The (x+y)/2 proof is simple, but the delta-proof also works for irrationals.
Which proof do you prefer? Why? What is your criteria?