TITLE: Prove that: $ (a<b<c) \implies (a<\frac{a+b+c}{3}<c) $
QUESTION [5 upvotes]: Prove that: $$ (a<b<c) \implies (a<\frac{a+b+c}{3}<c) $$
I'm having problem proving these implications (I don't know how they're called in English). Can you tell me what I have to read to understand these?

REPLY [7 votes]: HINT:
$a<b<c\implies 3a=a+a+a<a+b+c$
and $3c=c+c+c>c+b+a$