Moving from vague language and handwaving arguments to precise mathematical formulation is not easy. The first and most important step towards it are definitions of mathematical objects or properties. Definitions link common language with precise mathematical formulations.
For example, look at the definition of an even number:
Definition 1: An even number is an integer that is divisible by two.
Notice that to understand this definition we need to know the definition of the mathematical object `integer' and the definition of the mathematical property 'to be divisible by'. If we know these two definitions, we can reformulate the previous definition as:
Definition 2: A number z even if and only if z/2 ∈ ℤ.
You see the `if and only if' in the definition gives a logical equilavence between the expression "z is even" with the precise mathematical formulation "z/2 ∈ ℤ".
As you can see from this example, sometimes one has to rework a definition to get to the precise mathematical formulation, as we did when we went from definition 1 to definition 2.
Notice further, that one can even rewrite the definition in an equivalent, but slightly different way:
Definition 3: A number z is even if and only if there exists and integer k such that z=2k.
If you want to learn to write in a mathematical way, you need to start from the definitions.
Know definitions with precision. Learn how to express them mathematically (like going from Definition 1 above to Definition 2).
When solving exercises, for each mathematical object or property, you can only use their definitions. Anything else is not a mathematical argument.