TITLE: Proving $\sqrt{100,001}-\sqrt{100,000} < \frac{1}{2\sqrt{100,000}}$
QUESTION [5 upvotes]: Proving $\sqrt{100,001}-\sqrt{100,000} < \frac{1}{2\sqrt{100,000}}$
I squared both sides of the equation to get
$100,001 + 100,000+-200\sqrt{10}\sqrt{100,001} < \frac{1}{400,000}$.
I am just not sure how to justify this. I've tried multiplying both sides by -1, but it still would not hold.

REPLY [2 votes]: Another way:
$$\sqrt{a+1}-\sqrt a<\frac1{2\sqrt a}.$$
Multiplying by $2\sqrt a$ and moving the terms,
$$2\sqrt{(a+1)a}<1+2a.$$
Squaring,
$$4a^2+4a<1+4a+4a^2.$$
This works for any $a>0$ !