# Prove the sequences as in the Nested Interval Property.

Let $(x_n),(y_n)$ be sequences as in the Nested Interval Property.

Show that for all $n,m \in N$, $x_n \leq y_m$

We know from Nested Interval Property that

1. $x_1 \leq x_2 \leq x_3 \leq \cdots$

2. $y_1 \geq y_2 \geq y_3 \leq\cdots$

3. $x_n \leq y_n$ for all $n$

4. $\lim _{n\to\infty} (y_n-x_n) =0$

5. Squeeze Theorem

How can I use them to prove the question please?

• If $m < n$ then $x_m \leq x_n \leq y_n \leq y_m$. If $m > n$ then $x_n \leq x_m \leq y_m \leq y_n$. In both cases, $x_n \leq y_m$. – Bungo Mar 28 '17 at 21:17
• how about if m=n ? – qwer tyui Mar 28 '17 at 21:40
• If $m=n$ then simply observe that $x_n \leq y_n = y_m$. – Bungo Mar 28 '17 at 21:42
• I think before I had to use sequence definition to approach it but you give me easy way – qwer tyui Mar 28 '17 at 21:50