I have began working through Spivak's Calculus book and trying to do the problems at the end of the chapters. I am rather new to proof, so forgive the naivety of this type of question.
I am wanting to make sure that my method/ understanding of some of the proofs are correct, as I do not have solutions.
One I am wondering about for example is ,
$\mathbf{Q:}$ Prove that every natural number is even or odd.
$\mathbf{My \space answer:}$
First of all, let "A" be the set containing all natural numbers that are either even or odd. It is clear that $1 \in A$ because 1 is odd.
Now, by induction, suppose that "n" is in $A$.
Then n is either even , or n is odd
if $n$ is even, then it can be written as $2m$ where $m$ is an integer, and hence $n+1=2m+1$ hence $n+1$ is odd, and is in $A$.
If n is odd, then it can be written as $2m+1$ where $m$ is an integer, and hence $n+1=2m+2=2(m+1)$ and hence $n+1$ is even, so it is in $A$.
Is this sufficient to conclude then via induction that infact A is the set of all natural numbers?
Is there anything I am missing or should be careful about? Thanks everyone