I would like to give some definiton is that ; Let $S$ be a dominating subset of $[\mathbb{N}]^{\infty}$ and $S=\{s_{\alpha} : \alpha<\mathfrak{d}\}$ is called $\mathfrak{d}$-scale if for every $\alpha<\beta<\mathfrak{d}$ $s_{\beta}\nleq^{*} s_{\alpha}$.

In the paper from Tsaban, there is a lemma which is , 'every $\mathfrak{d}-scale$ is $\mathfrak{d}-concentrated$ on $[\mathbb{N}]^{<\infty}$.

for this lemma, how can we start? it is obvious that every $scale$ is $\mathfrak{d}-concentrated$ on $[\mathbb{N}]^{<\infty}$, but $\mathfrak{d}-scales$ any help will be appreciated. thank you in advance.

Let $S$ be a dominating subset of $[\mathbb{N}]^{\infty}$. Then $S=\{s_{\alpha} : \alpha<\mathfrak{d}\}$ is called a $\mathfrak{d}$-scale if for every $\alpha<\beta<\mathfrak{d}$, $s_{\beta}\nleq^{*} s_{\alpha}$. In the paper of Tsaban, there is a lemma: every $\mathfrak{d}$-scale is $\mathfrak{d}$-concentrated on $[\mathbb{N}]^{<\infty}$. How can we prove this lemma? It is obvious that every scale is $\mathfrak{d}$-concentrated on $[\mathbb{N}]^{<\infty}$, but every $\mathfrak{d}$-scale?