Direct Proof Writing Technique I've been having this proof writing hangup, and the following example covers the essence of it.   

In the beginning of my analysis book one of the exercises is: 

" If $a,b \in \mathbb{R}$ and $a + b = 0$, then $b = -a$. "

Here's the method I used -- and frequently use -- to prove it. I take the given information and derive the desired result. 

Proof
\begin{align}
a+b &amp;= 0 &amp;&amp;\text{given}\\
(-a)+a + b &amp;= (-a)+0  &amp;&amp;\text{definition of equality}\\
\left((-a)+a\right) + b &amp;= (-a)  &amp;&amp;\text{associativity of addition/zero element}\\
0+b &amp;=-a &amp;&amp;\text{additive inverse}\\
b &amp;= -a &amp;&amp;\text{zero element}\\
\end{align}

But the author uses this other (and much more common) proof style where you start from the left side of the desired result, use some given information, then get the right side. 

Proof

$$b = 0+b = (-a + a) + b = -a + (a+b) = -a + 0 = -a$$  

While I get the aesthetic, I just don't understand why it wouldn't be done the my way, which seems a lot easier. Is one of these styles more correct than the other, or preferred? And if so, why?  

P.S. I tagged this as a soft-question in case the answer is subjective.
The author's style has the benefit of explicitly exhibiting the axioms being used at each step. In fact that is probably the point of the exercise.
Your method leaves out many axioms, esp. in the last step which implicitly implies that $(a+b)-a = b  $ which needs something like commutativity, associativity, inverse, and identity axioms in that order, a sequence of four steps of the kind done by the author.
Your method is probably a good method of discovering the proof, but the author's method is better at showing the reasoning in this case. It is too bad that author's don't take the time to exhibit the difference.
If you claim that you don't need to go thru that axiomatic detail, then you really don't need to do this problem.
The author's style has the benefit of explicitly exhibiting the axioms being used at each step. In fact that is probably the point of the exercise.
Your method leaves out many axioms, esp. in the last step which implicitly implies that $(a+b)-a = b  $ which needs something like commutativity, associativity, inverse, and identity axioms in that order, a sequence of four steps of the kind done by the author.
Your method is probably a good method of discovering the proof, but the author's method is better at showing the reasoning in this case. It is too bad that author's don't take the time to exhibit the difference.
If you claim that you don't need to go thru that axiomatic detail, then you really don't need to do this problem.