I need the proof of squeeze lemma on sequences

[singedang2]

urgent! i need the proof of squeeze lemma on sequences

if [tex]y_n \leq x_n \leq z_n[/tex] and [tex]y_n \rightarrow p[/tex] and [tex]z_n \rightarrow p[/tex]

then [tex]x_n \rightarrow p[/tex]

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.

 

[StatusX]

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

 

[Thrice]

http://en.wikipedia.org/wiki/Squeeze_theorem

It's really easy though.

 

[singedang2]

how am i suppose to apply this into the sequences?

 

[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.

 

[HallsofIvy]

If [itex]a_n\le b_n\le c_n[/itex] and [itex]\lim a_n= \lim c_n= L[/itex] then [itex]lim b_n= L[/itex].

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