First of all I do not know if this even qualifies to be a proper question, but I have a rather trivial doubt which I somehow could not resolve for the past few hours. So here it is.
Let $A\in \mathbb{R}^{m\times N}$ be a random matrix which satisfies some concentration inequality of the form $$P\left(\left|\|Ax\|^2-E(\|Ax\|^2)\right|\ge \delta\right)\le 2\exp(-cf(\delta,m))\quad \forall x\in \mathbb{R}^N$$ for some $1>\delta>0$ and some function $f:\mathbb{R}\to \mathbb{R}$.
From here it seems to me that this statement implies that $$P(\sigma_{max}(A)\le \sqrt{1+\delta})\ge 1-\exp(-cf(\delta,m))\\ P(\sigma_{min}(A)\ge \sqrt{1-\delta})\ge 1-\exp(-cf(\delta,m))$$ But I am not sure about it and in fact have seen papers where they talk about methods like Slepin-Gordon to use to find the extremal value probability bound. I think I am missing something. I suspect I am not considering the fact that for getting the tail bounds the former inequalities have to be satisfied $for\ all \ x\in \mathbb{R}^n$. Though this seems to me quite a trivial issue but it is bothering me constantly. So my question is, is my doubt correct or something more sinister is going on here? Thanks in advance.