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I need the proof of squeeze lemma on sequences

  1. Oct 19, 2006 #1
    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.
     
  2. jcsd
  3. Oct 19, 2006 #2

    StatusX

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    What have you tried? You'll need to use the epsilon delta definition of the limit.
     
  4. Oct 19, 2006 #3
  5. Oct 19, 2006 #4
    how am i suppose to apply this into the sequences?
     
  6. Oct 19, 2006 #5

    StatusX

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    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.
     
  7. Oct 20, 2006 #6

    HallsofIvy

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    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]?
     
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