Strong monotonicity and weak monotonicity
Let $x,y\in X$. Suppose $y \gg x$ (so that in particular, $y\geq x$ and $y\neq x$). By assumption, $\succsim$ satisfies strong monotonicity; and so, $y\succ x$.

(We've just shown that for any $x,y\in X$ such that $y \gg x$, we have $y\succ x$. Hence, $\succsim$ satisfies weak monotonicity.)



The argument you've given above is correct and roughly the same as the proof I've just given. The only problem is that your argument is somewhat indirect, convoluted, and unclear.