(Algebra) I'm looking at a solution to an Intro Real Analysis problem, but there's a simple algebraic step being made in the solution that I don't get. It seems like they're multiplying by the conjugate of something, but not quite. Prompt below.

Question

mendicant_bias · Answer

Problem one on page one: http://www.math.cornell.edu/~mionescu/Teaching/Spring2008/prelim2_311_sol.pdf I don't understand what is happening when they get the expression of n's in absolute values that is less than epsilon that becomes the conjugate and the reciprocal of what it previously was?

mendicant_bias · Answer

I'd like to say they just multiplied (a-b) time (a+b) on top and bottom, but I don't see how that would lead to the given result.

mendicant_bias · Answer

@freckles

thomas5267 · Answer

$$
\sqrt{n+1}-\sqrt{n}<\epsilon\
\left(\sqrt{n+1}-\sqrt{n}\right)\left(\sqrt{n+1}+\sqrt{n}\right)<\epsilon\left(\sqrt{n+1}+\sqrt{n}\right)\
n+1-n<\epsilon\left(\sqrt{n+1}+\sqrt{n}\right)\
1<\epsilon\left(\sqrt{n+1}+\sqrt{n}\right)\
\frac{1}{\sqrt{n+1}+\sqrt{n}}<\epsilon
$$

thomas5267 · Answer

$$
(a+b)(a-b)=a^2-b^2
$$

thomas5267 · Answer

And this does not look like an introductory course to analysis.