Prove Limit of Sequence: Sqrt(n+1)-Sqrt(n) = 0

[tarheelborn]

Homework Statement

Prove that the limit of the sequence {Sqrt(n+1)-Sqrt(n)} = 0.

Homework Equations

The Attempt at a Solution

I know that I must multiply by the conjugate to come up with 1/(Sqrt[n+1]-Sqrt[n]) and that the limit of this is clearly 0. I am having trouble solving this equation in terms of epsilon.

 

[Tedjn]

Be careful of the sign; multiplying by the conjugate gives you 1/{sqrt(n+1) + sqrt(n)}. For all ε > 0, there exists 1/m < ε for some positive integer m. (Why?) How should you choose N so that 1/{sqrt(n+1) + sqrt(n)} < 1/m < ε whenever n > N?

 

[tarheelborn]

Ah, yes, the sign. Thank you! Now... To choose N, can I ignore the sqrt(n) part of the denominator since sqrt(n+1) > sqrt(n) so, being in the denomominator, the number is smaller? So could I let N = (1/epsilon^2) - 1?

 

[Tedjn]

That should work fine; by convention we let N be an integer, so you can set N to be the ceiling of what you have.

 

[tarheelborn]

Thank you!