Show that the order of deletion doesn’t matter for the set of strategies surviving iterated elimination of strictly dominated strategies

Question

Question from M.W.G-8B4

Show that the order of deletion doesn’t matter for the set of strategies surviving iterated elimination of strictly dominated strategies.

My proof attempt

I assume $G_n$ the game generated in the n^th step of IDDs.

Suppose $s_i^D \in \Sigma_i^n $ dominates $s_i\in \Sigma_i^n$ for given $\Sigma_{-i}^n $

But let’s assume it is not deleted in this round.

Then at round $n+1$, $s_i^D \in \Sigma_i^{n+1} $ still dominates $s_i\in \Sigma_i^{n+1} $ for given $\Sigma_{-i}^{n+1} $

So, the order of deletion doesn’t matter.

— what do you think about my proof attempt? Is this enough to show this statement? Or how can improve this proof?

Any help will be appreciated. Thanks!