# I need the proof of squeeze lemma on sequences

1. Oct 19, 2006

### singedang2

urgent!! i need the proof of squeeze lemma on sequences

if $$y_n \leq x_n \leq z_n$$ and $$y_n \rightarrow p$$ and $$z_n \rightarrow p$$

then $$x_n \rightarrow p$$

Note. i'm not looking for the proof of the regular squeeze theorem. this is supposed to be a proof adapting the proof of squeeze theorem onto the sequences.

2. Oct 19, 2006

### StatusX

What have you tried? You'll need to use the epsilon delta definition of the limit.

3. Oct 19, 2006

### Thrice

4. Oct 19, 2006

### singedang2

how am i suppose to apply this into the sequences?

5. Oct 19, 2006

### StatusX

Do you know the definition of the limit of a sequence? It's very similar in form to the epsilon delta definition used for functions.

6. Oct 20, 2006

### HallsofIvy

Staff Emeritus
If $a_n\le b_n\le c_n$ and $\lim a_n= \lim c_n= L$ then $lim b_n= L$.

Since $lim a_n= L$, then, given any $\epsilon$ for some N1, if n> N1, $|a_n- L|< \epsilon$. Since $lim c_n= L$, given any $\epsilon$ for some N2, if n> N2, $|c_n- L|< \epsilon$. If n> larger of (N1, N2) what can you say about both $a_n$ and $c_n$. What does that tell you about $c_n$?