For a natural number n and real numbers a1, a2,...,an, verify that:
|a1 + a2 + ... + an| <= sqrt(n)*sqrt(a12 + a22 + ... + an2)
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.