Ok, so I have the following to prove:
If a and b are real numbers s.t. 0<a<b, then there is a natural number n s.t. a<b-1/n<b.
I am confused as to what I should do to prove this. Is this a two part proof? If so, what parts do I prove? Thanks.
Ok, so I have the following to prove:
If a and b are real numbers s.t. 0<a<b, then there is a natural number n s.t. a<b-1/n<b.
I am confused as to what I should do to prove this. Is this a two part proof? If so, what parts do I prove? Thanks.
Consider the set of natural numbers
This is a non-empty set of natural numbers that is bounded from below.
So by the well ordering of the postive integers this set has a least element
Obviously
Now we just need to show that
we know that
Putting the above two inequalities together we get