Technical question regarding showing sqrt(n+1) - sqrt(n) converges to 0

[gmn]

Homework Statement

show the sequence s_n= (n+1)^1/2 - n^1/2 converges to zero

Homework Equations

The Attempt at a Solution

I don't have that much of a problem showing the limit goes to zero, rationalize the numerator (or whatever it's called) to get (n+1)^1/2 - n^1/2 = 1/((n+1)^1/2 + n^1/2). My question is that I show this goes to zero because s_n<1/n(^1/2) which goes to zero, but my professor provides a solution where he writes s_n<1/2(n^1/2). I don't understand why the 2 is there. Is saying that s_n<1/(n^1/2) insufficient or not true?

Thanks

 

[lanedance]

both are true, and sufficient to show it converges to zero, the 2nd is just a little tighter

how about this, as
$$(n+1)^{1/2} > n^{1/2}$$
then
$$\frac{1}{s_n} = n^{1/2} + (n+1)^{1/2} > n^{1/2} + n^{1/2} = 2n^{1/2}$$
then inverting
$$s_n = \frac{1}{n^{1/2} + (n+1)^{1/2}} < \frac{1}{n^{1/2} + n^{1/2}} = \frac{1}{2n^{1/2}}$$

 

[dalle]

$$\sqrt{n+1}+\sqrt{n}> 2 \sqrt{n}$$.
taking the inverse on both sides yields
$$\frac{1}{\sqrt{n+1}+\sqrt{n}} < \frac{1}{2 \sqrt{n}}$$
your professor is just using a smaller upper bound for $$s_n$$. professors like to use bounds that are as small as possible