- #1

- 19

- 0

**1. Homework Statement**

Prove that if x is any positive integer, then ⌈x/2⌉ ≤ (x + 1)/2. (Here, for

any real number r, ⌈r⌉ is the smallest integer greater than or equal to r. Thus, ⌈2.3⌉ = 3,

⌈2⌉ = 2, etc.) Do a proof by cases.

**2. Homework Equations**

None that I know of.

**3. The Attempt at a Solution**

Since it's a proof by cases, and it only deals with positive integers, I think my two cases should be if x is an even integer, and an odd integer. This is the point where I get confused, do I basically plug in 2k and 2k+1 for x, and then solve? If that is true? What exactly am I solving for?

Anyways for the next proof:

**1. Homework Statement**

Show that if r is an irrational number, there is a unique integer n such that the distance between r and n is less than 1/2. Be sure to argue the uniqueness of n.

**2. Homework Equations**

None that I know of.

**3. The Attempt at a Solution**

I have absolutely no idea on where to even start for this problem. I all know is that since r is irrational, it can be said that it could lie in between two integers n and n +1, much like n < r < n + 1. So I have to solve for n < r and r < n + 1. But again, what exactly am I solving for, and what do I have to do to show that? I guess my main problem in these types of proofs is just verifying they the cases work, I really just dont see how any of the cases could lead to a conclusion that I could use to help prove the problems. Anyways, any help is appreciated, thanks in advance!