Hi, im not sure if this is in calculus or number theory section...
Using the definition of a convergent sequence show that$\displaystyle
\sqrt{n+1} - \sqrt{n}
$ converges to zero.
Thanks in advance!!
-ynot-
thanks, do i do this?
$\displaystyle \frac 1{2\sqrt{n}}<epsilon$
then n>1/(4epsilon^2)
thus |An - 0| < epsilon if n>1 & n>1/(4epsilon^2)
n*> max{1,1/(4epsilon^2)}
n>=n*, we have |An-0|<epsilon
[SORRY it's taking me forever to type it in the [math} mode]
you want to show that
For every $\displaystyle \epsilon > 0$, there exists an $\displaystyle N \in \mathbb{N}$, such that $\displaystyle n > N$ implies
$\displaystyle |\sqrt{n + 1} - \sqrt{n} - 0| < \epsilon$
So let $\displaystyle \epsilon > 0$ and choose $\displaystyle N$ such that $\displaystyle N > \frac 1{4 \epsilon ^2}$
Then blah blah blah $\displaystyle < \epsilon$
QED