If $M\subset N \Rightarrow N^{\bot}\subset M^{\bot}$. Seems too easy to be right.

Your idea is entirely correct, but the way you write it is a little weird. When you write

$$\langle u, v'\rangle=\langle u,v\rangle,$$

at this point, you have not yet defined what $v$ is. It is better to just write it like so:

If $v'\in M$, then we know that $v'\in N$ because $M\subset N$. Because $\langle u,v\rangle = 0$ for all $v\in N$, it is also true for $v=v'$, which means $\langle u, v'\rangle = 0$.

Or, you can write it even shorter, simply as

If $v'\in M$, then $v'\in N$ because $M\subset N$. By definition, because $u\in N^\bot$ and $v'\in N$, we have $\langle u, v'\rangle = 0$.