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Analysis: Sequence convergence with Square Roots

  1. Jan 25, 2008 #1
    1. The problem statement, all variables and given/known data
    Given Lim Cn=c, Prove that Lim[tex]\sqrt{Cn}[/tex]=[tex]\sqrt{c}[/tex]


    2. Relevant equations
    We are working from the formal definition: for all [tex]\epsilon[/tex], there exists an index N such that For all n>=N, |Cn-c|<[tex]\epsilon[/tex]


    3. The attempt at a solution
    We as a group have attempted this several times from several directions, too many to post here. we've been working for over an hour with no results. Please HElp!!!
     
  2. jcsd
  3. Jan 25, 2008 #2
    So you can't use any limit theorems?
     
  4. Jan 25, 2008 #3
    Let [itex]\epsilon_1>0\Rightarrow \epsilon_1\,(\epsilon_1+2\sqrt{c})>0[/itex] then

    [tex]\exists \,N:|C_n-c|<\epsilon_1\,(\epsilon_1+2\sqrt{c}), \forall n\geq N [/tex]

    Thus [itex] \forall\, n\geq N[/itex]

    [tex]C_n-c<\epsilon_1\,(\epsilon_1+2\sqrt{c}) \Rightarrow C_n<\epsilon_1^2+2\,\epsilon_1\,\sqrt{c}+c\Rightarrow C_n<(\epsilon_1+\sqrt{c})^2\Rightarrow \sqrt{C_n}<\epsilon_1+\sqrt{c}[/tex]

    Now choose [itex]\epsilon=max\Big(\epsilon_1\,(\epsilon_1+2\sqrt{c}),\epsilon_1+2\sqrt{c}\Big)[/itex] and apply the triangle inequality at [itex]|\sqrt{C_n}-\sqrt{c}|[/itex].
     
  5. Jan 25, 2008 #4
    [itex] |c_n -c | = | \sqrt{c_n} - \sqrt{c}| |\sqrt{c_n} +\sqrt{c}| [/tex] should work i think . you can prove [tex]|\sqrt{c_n} +\sqrt{c}| [/itex] is bounded using the definition you gave in 2 and choose a new epsilon
    Of course i am assuming we are talking about a nonegative sequence
     
    Last edited: Jan 25, 2008
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