Let $1 \le n < N$ be integers and $A$ be a random $N\times n$ matrix with iid entries from $\mathcal N(0,1)$. This paper (Rudelson and Vershynin) claims in the paragraph just before formula (3.4) that, there exists a constant $c>0$ such that

$$ P(s_\min(A) \ge c\sqrt{N}) \ge 1 - 2e^{-CN},\text{ with }C=1! \tag{1} $$

I doubt the validity of the above statement with $C=1$.

Note. I know that the statement can be made true with a smaller value of $C$ (and some mild conditions on the aspect ratio $n/N$).

Question. Can someone kindly rollout the sketch of a proof of (1), or else disprove it ?

Thanks in advance!