Analysis: Sequence convergence with Square Roots

[clifsportland]

Homework Statement

Given Lim Cn=c, Prove that Lim\sqrt{Cn}=\sqrt{c}

Homework Equations

We are working from the formal definition: for all \epsilon, there exists an index N such that For all n>=N, |Cn-c|<\epsilon

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!

 

[Mathdope]

So you can't use any limit theorems?

 

[Rainbow Child]

Let \epsilon_1&gt;0\Rightarrow \epsilon_1\,(\epsilon_1+2\sqrt{c})&gt;0 then

\exists \,N:|C_n-c|&lt;\epsilon_1\,(\epsilon_1+2\sqrt{c}), \forall n\geq N

Thus \forall\, n\geq N

C_n-c&lt;\epsilon_1\,(\epsilon_1+2\sqrt{c}) \Rightarrow C_n&lt;\epsilon_1^2+2\,\epsilon_1\,\sqrt{c}+c\Rightarrow C_n&lt;(\epsilon_1+\sqrt{c})^2\Rightarrow \sqrt{C_n}&lt;\epsilon_1+\sqrt{c}

Now choose \epsilon=max\Big(\epsilon_1\,(\epsilon_1+2\sqrt{c}),\epsilon_1+2\sqrt{c}\Big) and apply the triangle inequality at |\sqrt{C_n}-\sqrt{c}|.

 

[matness]

|c_n -c | = | \sqrt{c_n} - \sqrt{c}| |\sqrt{c_n} +\sqrt{c}| [/tex] should work i think . you can prove |\sqrt{c_n} +\sqrt{c}|is bounded using the definition you gave in 2 and choose a new epsilon<br /> Of course i am assuming we are talking about a nonegative sequence