1. ## Pigeonhole Problem

Let n be a positive integer and let a1, a2,...,a(n+1) be an real numbers in the interval [0,1). Show that there exists two integers i,j with 1< i, j < (n+1) and i doesn't equal j such that |ai - aj| < (1/n).

Ok. I can come up with an example...ai = .25 and aj = .3 and n = 5, but I'm sure that it's asking for a more general answer...

Anyone have any ideas?

2. Express the interval $\displaystyle [0,1)$ as the union of $\displaystyle n$ subintervals.

$\displaystyle [0,1)=[0,1/n) \cup [1/n,2/n) \cup \dots \cup [(n-1)/n, 1)$.

Since there are $\displaystyle n+1$ of the $\displaystyle a_i's$, at least two of them must be in one of the $\displaystyle n$ intervals; i.e,

$\displaystyle \exists i,j$ s.t $\displaystyle i\not= j$ and $\displaystyle a_i,a_j \in [k/n, (k+1)/n)$, for $\displaystyle 0\le k \le n-1$.

Therefore $\displaystyle |a_i-a_j| < 1/n$.

3. Originally Posted by Black
Express the interval $\displaystyle [0,1)$ as the union of $\displaystyle n$ subintervals.

$\displaystyle [0,1)=[0,1/n) \cup [1/n,2/n) \cup \dots \cup [(n-1)/n, 1)$.

Since there are $\displaystyle n+1$ of the $\displaystyle a_i's$, at least two of them must be in one of the $\displaystyle n$ intervals; i.e,

$\displaystyle \exists i,j$ s.t $\displaystyle i\not= j$ and $\displaystyle a_i,a_j \in [k/n, (k+1)/n)$, for $\displaystyle 0\le k \le n-1$.

Therefore $\displaystyle |a_i-a_j| < 1/n$.
hey, can you explain that in little easier language if there is any!!!
thanksss