Synopsis:

Several tactics may be used to write mathematical proofs, usage of theorem prover is also increasingly common. (Although quite rough.)
The most widely used method of writing proofs is pen and paper alongside peer review. One may state the following

There exist solutions consisting of two natural numbers a, and b: \[ a + b = c. \] Where \( c \neq 0 \land a + b < c \land \forall a, b \in \mathbb{N}^{+} \)

Such trivial looking proofs is actually a dumbed-down version of the ABC conjecture (or theorem if you live in Japan.), one of the most non-trivial conjectures you can attack.

What can I learn then?

When writing proofs, one shall remember that every step must be justified and explicitly written. A good example of such proofs would be in Number Theory and Algebraic Topology.