Confirming a easy proof: the product of two consecutive numbers is always even. Can someone confirm if my prove is right?
Theorem. The product of two consecutive integers is always even.
Proof. Define a number $n$ such that $n:=2k$ where $k\in\mathbb{Z}$, this ensures that $n$ is an even number. Define a second number $p$ such that $p:=n+1=2k+1$ this ensures that $p$ is an odd number and the numbers $n$ and $p$ are consecutive because they differ by $1$. There product is given by:
$$np=2k(2k+1)=2(2k^2+k)=4k^2+2k\tag1$$
Which is clearly a multiple of $2$.
Define a number $h$ such that $h:=2m+1$ where $m\in\mathbb{Z}$, this ensures that $h$ is an odd number. Define a second number $z$ such that $z:=n-1=2m$ this ensures that $z$ is an even number and the numbers $h$ and $z$ are consecutive because they differ by $1$. There product is given by:
$$hz=2m(2m+1)=2(2m^2+m)=4m^2+2m\tag2$$
Which is clearly a multiple of $2$, this proves that $hz$ is an even number.
This proves this theorem.

If I am wrong how can I make my proof valid?


You proved that the product of an even number and the next number is even.
Since any number is even or odd, you could complete the proof of the theorem
by proving that the product of an odd number and the next number is also even.
