- #1

t00dles23

- 13

- 0

## Homework Statement

For a natural number n and real numbers a

_{1}, a

_{2},...,a

_{n}, verify that:

|a

_{1}+ a

_{2}+ ... + a

_{n}| <= sqrt(n)*sqrt(a

_{1}

^{2}+ a

_{2}

^{2}+ ... + a

_{n}

^{2})

## Homework Equations

I suspect that this can be done using properties of the inner product (i.e. the Cauchy-Schwarz inequality), or the triangle inequality, but I just can't seem to make it come out.

## The Attempt at a Solution

It obviously would be sufficient to prove that (a_1 + a_2 + ... + a_n)^2 <= n(a_1)^2 + n(a_2)^2 + ... + n(a_n)^2. But try as I might I can't figure a strategy to show this. The square of an n term sum of numbers is by no means pretty, and I don't have a good formula for it.