Keeping the distance

Algebra Level 5

$\large\left|\frac{a_{1}b_{1} + a_{2}b_{2} + ... + a_{n}b_{n}}{\sqrt{a_{1}^2 + a_{2}^2 + ... + a_{n}^2}} \right| - \sqrt{ b_{1}^2 + b_{2}^2 + ... + b_{n}^2} > 0$ .

Let $a_1,a_2,\ldots,a_n$ and $b_1,b_2,\ldots,b_n$ be real numbers for which the inequality above is fulfilled. Which of the answer choices given is true?

See my set of Problems.

a_{1} + a_{2} + \ldots+ a_{n} < -1

None of these choices

a_{1} + a_{2} +\ldots+ a_{n} < 1

a_{1} + a_{2} + \ldots+ a_{n} < 0

a_{1} + a_{2} + \ldots+ a_{n} < -2

a_{1} + a_{2} + \ldots + a_{n} < 2