When you are given information and asked to show 'something' is it completely wrong to start with that 'something' and work back to the information?
Can you start with what you want to show, work backwards a few steps using equivalence and then show that.For example I was told i cant work backwards in the problem:
Let $a,b≥2$ Show that $a+b≤ab$. What i did which was wrong was start with what i had to show then reached $(a-1)(b-1)≥1$ which i said was satisified by $a,b≥2$. This is backwards and wrong,
I know you should start with $a+b≤ab$ show equivalence to $a+b-ab≤0$ and then i think the problem reduces to showing $a+b-ab$ is greater than 0 but its quite confusing to me if i am working backwards here or not because if you start from $a,b>2$ and go to $a+b-ab$ it seems as though you have.
Thanks
To show that a certain statement $P$ is true, you can work backwards. However, you need to make sure that every step is reversible, meaning, if $P\implies S$, then $S\implies P$ as well. In your example, you have $$a+b\leq ab \iff ab-a-b+1\geq1 \iff (a-1)(b-1)\geq 1$$ which is true for the given conditions. Now you can start from the last statement, which is true, and go in the reverse direction as permitted by the direction of the arrows to reach the statement you are trying to prove. This way is valid. However, if you only had $$a+b\leq ab \implies ab-a-b+1\geq1 \implies (a-1)(b-1)\geq 1$$ where at least one arrow only points in the forward direction, then even if you reach a correct conclusion, you cannot say that the initial statement was true. This is because your logic doesn't work the other way around anymore. For example, $$0\geq1 \implies 0\cdot0\geq0\cdot1 \iff 0\geq0$$ which is true, but since the first arrow only points forward, the initial statement need not be true (indeed, it's false in this case).