Small advice @ mulferlease learn LATEX

We have to prove:

It is equivalent to proving:

So we are done if we are able to prove this.

now,

if is a positive real number,

if then (obvious)

if then (obvious)

if then so, (multiplying both sides by a negative number i.e changes the sign)

so for all case,

similarly,

summing these we get:

so we will be done if we prove or

by the AM-GM inequality,

# as

proved