I am a student and I want to know if this proof is correct.

**Proof.** If $\sqrt5$ is a natural number, it should be even or odd.
If it would be even, we could express it as $2k$ for some integer $k$. 
If it would be odd, we could express it as $2j+1$ for some integer $j$. 
But those integers do not exist what is a contradiction. 
Then, $\sqrt5$ is not a natural number. Q.E.D.

Using Joofan's idea, I have a second proof. Please give me your opinion!

**Proof.** If $\sqrt5$ is a natural number, then there exists a natural number $n^2=5$, but $2^2<5<3^2$, and there is no natural number between $2$ and $3$. Therefore, we can conclude that there does not exist a natural number $n^2=5$, and that is a contradiction. Q.E.D.