Question: Given a square random matrix $X\in\mathbb{R}^{p\times p}$ where $$ X_{ij}\stackrel{iid}{\sim}\text{Bernoulli}(\alpha), $$ where $\alpha\in(0,1)$, satisfies the condition $$ \|X\|_{\text{op}}<C \text{ almost surely}, $$ where $\|X\|_{\text{op}}$ is the operator norm of the matrix (see definition 10.2 of this note) and $C$ is some constant. Can I use this result to show that for a rectangular matrix $\widetilde{X}\in\{0,1\}^{n\times p}$ where $n<p$, with entries generated in the same manner, the same condition $\|\widetilde{X}\|_{\text{op}}<C$ holds? Note that the only difference is that we shifted from a square matrix to a fat-rectangular matrix (with no other changes). Thanks.
Attempt: I can argue that the result also holds for $\widetilde{X}$ because we can always pad $\widetilde{X}$ with more i.i.d. samples to obtain a square matrix that satisfies the bound almost surely. However, I find my argument to be not very rigorous... Is there a more rigorous argument, or a way to make my argument more rigorous? Thanks.