MathDB

Prove the inequality
a.)

\left( a_{1}+a_{2}+...+a_{k}\right) ^{2}\leq k\left( a_{1}^{2}+a_{2}^{2}+...+a_{k}^{2}\right) ,

where

k\geq 1

is a natural number and

a_{1},

a_{2},

...,

a_{k}

are arbitrary real numbers.
b.) Using the inequality (1), show that if the real numbers

a_{1},

a_{2},

...,

a_{n}

satisfy the inequality

a_{1}+a_{2}+...+a_{n}\geq \sqrt{\left( n-1\right) \left( a_{1}^{2}+a_{2}^{2}+...+a_{n}^{2}\right) },

then all of these numbers

a_{1},

a_{2},

\ldots,

a_{n}

are non-negative.

Problem Statement