Utility Guarantee of Small Data Base Mechanism in Differential Privacy

I am reading Section 4.1 (An offline algorithm: SmallDB) of The Algorithmic Foundations of Differential Privacy by Dwork and Roth. I am stuck at the proof of Proposition 4.4, which is about the utility guarantee of the small database mechanism (Algorithm 4 in page 70).

Proposition 4.4. Let $$\mathcal{Q}$$ be any class of linear queries. Let $$y$$ be the database output by SmallDB($$x,\mathcal{Q},\epsilon,\alpha$$). Then with probability $$1-\beta$$: $$\max_{f\in\mathcal{Q}}|f(x)-f(y)|\le\alpha+\frac{2\left(\frac{\log|\mathcal{X}|\log|\mathcal{Q}|}{\alpha^2}+\log\left(\frac{1}{\beta}\right)\right)}{\epsilon\|x\|_1}.$$ Proof. Applying the utility bounds for the exponential mechanism (Theorem 3.11) with $$\Delta u=\frac{1}{\|x\|_1}$$ and $$\text{OPT}_q(x)\le\alpha$$ (which follows from Theorem 4.2), we find: $$\Pr\left[\max_{f\in\mathcal{Q}}|f(x)-f(y)|\ge\alpha+\frac{2}{\epsilon\|x\|_1}\left(\log|\mathcal{R}|+t\right)\right]\le e^{-t}.$$

Here $$\mathcal{R}=\{y\in\mathbb{N}^{|\mathcal{X}|}:\|y\|_1=\log|\mathcal{Q}|/\alpha^2\}$$ is the collection of small databases, and $$u:\mathbb{N}^{|\mathcal{X}|}\times\mathcal{R}\to\mathbb{R}$$ is given by $$u(x,y)=-\max_{f\in\mathcal{Q}}|f(x)-f(y)|,$$ which measures the similarity between an arbitrary database $$x$$ and a small database $$y\in\mathcal{R}$$.

My question: Why is $$\Delta u=1/\|x\|_1$$? According to page 38, $$\Delta u$$ is defined by $$\Delta u=\max_{y\in\mathcal{R}}\max_{\|x-x'\|_1\le 1}|u(x,y)-u(x',y)|,$$ so $$\Delta u$$ should not depend on the size of any database $$x$$.

$$f(x) = \dfrac{1}{||x||_1}\sum_{i=1}^{|\mathcal{X}|}x_i f(\mathcal{X}_i).$$
In this way, by using the neighboring definition of one different user (so the neighboring databases have the same size $$||x||_1$$ but they differ by one row), the utility function $$u(x,y)$$ takes the normalization factor in its global sensitivity from the function $$f$$.
$$\Delta u = \max_{y\in \mathcal{R}}\max_{||x-x'||_1\leq 1}|u(x,y)-u(x',y)| = \max_{y\in \mathcal{R}}\max_{||x-x'||_1\leq 1}\dfrac{1}{||x||_1}|\tilde{u}(x,y)-\tilde{u}(x',y)| =\dfrac{1}{||x||_1},$$
where $$\tilde{u}$$ is the un-normalized version of $$u$$, hence a counting queries with sensitivity 1.