In the following $|.|$ denotes the operator norm in the space $GL(n):=$set of invertible matrix of size $n \times n$.
I have the next problem: Let $S \in GL(n)$ and $\mathcal{G}$ be a dense subset of $GL(n)$, given $t>1$ and $\varepsilon$ with $\frac{1}{t} + \varepsilon < 1 < t -\varepsilon$ show that there exist $T \in \mathcal{G}$ such that $$|ST^{-1}| \leq t-\varepsilon \text{ and } |TS^{-1}| \leq \left(\frac{1}{t}+\varepsilon \right)^{-1} $$ First, i try to use the next fact:
If $|S-T| \leq \delta$ then $|ST^{-1}| \leq 1+\delta|T^{-1}|$ and $|TS^{-1}| \leq 1+\delta|S^{-1}|$
Then i solve for $\delta$
$$1+\delta|T^{-1}|=t-\varepsilon \text{ and } \left(\frac{1}{t}+\varepsilon \right)^{-1}=1+\delta|S^{-1}|$$
Finding 2 diferent solutions $\delta_1=\frac{t-\varepsilon-1}{|T^{-1}|}$ and $\delta_2=\frac{1-\frac{1}{t}-\varepsilon}{\frac{|S^{-1}|}{t}+\varepsilon|S^{-1}|}$, then if define $\delta:=\min\{\delta_1, \delta_2\}$ its clear that $\delta_1$ depends of $T$ and i think to this is not correct, i try to bound $|T^{-1}|$ in terms of $|T-S|$ but i can´t conclude.
Any hint or help i will be very grateful.
Thanks!