An operator $T:X\rightarrow Y$ is said to be upper semi-Fredholm if its range is closed and its kernel is finite-dimensional. M. Schechter (1972) introduced a quantity $$\nu(T):=\sup_{\operatorname{codim} M<\infty}\inf_{x\in M, \,\|x\|=1}\|Tx\|$$ and proved that $T$ is upper semi-Fredholm if and only if $\nu(T)>0$.
For a bounded subset $A$ of a Banach space $X$, let $\chi(A):=\inf\{\epsilon>0: A$ has a finite $\epsilon$-net in $X\}$. Then $A$ is relatively norm-compact if and only if $\chi(A)=0$. For an operator $T:X\rightarrow Y$, let $$h_{\mathrm{cb}}(T):=\inf\{\chi(TD):\chi(D)=1\},$$ where the infimum is taken over all countable bounded subsets $D$ of $X$ with $\chi(D)=1$.
M. González and A. Martinón (1995) noted that these two quantities are equivalent:
$$\frac{1}{2}h_{\mathrm{cb}}\leq \nu\leq 2h_{\mathrm{cb}}.$$
But they did not provide the proof. I need a detailed proof of this result.
