The discussion revolves around proving mathematical inequalities without using specific methods such as the AM-GM inequality or contradiction. Participants are exploring the inequality involving three variables, \(a\), \(b\), and \(c\), and also discussing the implications of the squeeze theorem in this context.
The discussion remains unresolved, with multiple competing views on the proofs and the relevance of the squeeze theorem. Participants are seeking clarification and further explanation.
Participants have not reached consensus on the proofs or the application of the squeeze theorem, and there are unresolved questions regarding the connections between the concepts discussed.
my solution:solakis said:1)Prove without using AM-GM :$$\frac{ab}{c}+\frac{ac}{b}+\frac{bc}{a}\geq a+b+c$$...... a,b,c >02) Prove without using contradiction :
$$a\leq b\wedge b\leq a\Longrightarrow a=b$$
Albert said:my solution:
(1)$\times abc$
we have to prove :$(ab)^2+(bc)^2+(ca)^2>(abc)\times (a+b+c)=a^2bc+b^2ca+c^2ab$
for $(A-B)^2=A^2-2AB+B^2$
$\therefore A^2+B^2>2AB---(3)$
likewise $B^2+C^2>2BC---(4)$
$C^2+A^2>2CA---(5)$
(3)+(4)+(5) we have $A^2+B^2+C^2>AB+BC+CA$
let $A=ab,B=bc,C=ca$
and the proof is done
(2)The statement A ∧ B is true if A and B are both true; else it is false.(definition of ∧:logical conjunction)
by using squeeze theorem we get:
$$a\leq b\wedge b\leq a\Longrightarrow a=b$$
squeeze theorem:solakis said:Sorry , but what is the squeeze theorem ??
Albert said:squeeze theorem:
https://en.wikipedia.org/wiki/Squeeze_theorem
solakis said:I am sorry but i fail to see the connection.
If you would care to elaborate a bit more