hi, ive recently started looking into maths outside of class and am mostly interested in proof. however, when reading into it a bit i got confused as to what the book meant by a true proof, and a non-proof, and i would like you to clear a few things up for me if you can :D thnx. this is the non-proof: Prove that [tex]\surd2 + \surd6 < \surd15[/tex] [tex]\surd2 + \surd6 < \surd15 \Longrightarrow (\surd2 +\surd6)^2 < 15 \Longrightarrow 8 + 2\surd12 < 15 \Longrightarrow 2\surd12 < 7 \Longrightarrow 48 < 49[/tex] is says the implication is going the wrong way. so this is the real proof: Prove that [tex]\surd2 + \surd6 < \surd15[/tex] [tex]\surd2 + \surd6 \geq \surd15 \Longrightarrow (\surd2 +\surd6)^2 \geq 15 \Longrightarrow 8 + 2\surd12 \geq 15 \Longrightarrow 2\surd12 \geq 7 \Longrightarrow 48 \geq 49[/tex] now i can just about understand how that works. because the implication thing i mean. for example, if M implies N it shows nothing of the truth of N. so, we have to form the negation of [tex]\surd2 + \surd6 < \surd15[/tex], which is [tex]\surd2 + \surd6 \geq \surd15[/tex], and then prove that there is a contradiction. What that seems to imply to my nooby brain is that all proofs using inequalities need to be proven by proof by contradiction. Is this right? And then is this the same with equalities? so does this proof work? prove [tex]2^3 + 2*3 = 2*7[/tex]: [tex]2^3 + 2*3 = 2*7 \Longrightarrow 8 + 6 = 14 \Longrightarrow 14=14[/tex] so i havn't proved that one with the equals using proof by contradiction, but then it seems to make sense to me? what is the difference with inequalities? Am I totally confused? Am I totally wrong? can someone please help. thnx p.s. excuse my noobishness, i just want to get things right from the get go lol.