Let $\{c_{i}\}_{i\in\{1,\dots,m\}}\subset\mathcal{C}$ be a finite set of generators of $\mathcal{C}$ over $\mathcal{A}$, then consider an $\mathcal{A}$-Tate algebra $\mathcal{A}\{\boldsymbol{r}^{-1}\boldsymbol{T}\}$ where $r_{i}\geq\vvvert c_{i}\vvvert_{\mathcal{C}}$. There is a surjective $k$-algebra homomorphism defined by

$$\gamma:\mathcal{A}\{\boldsymbol{r}^{-1}\boldsymbol{T}\}\to\mathcal{C},\text{ }T_{i}\mapsto c_{i}$$

which is bounded as there exists $C>0$ such that

$$\vvvert\gamma(\sum_{J}a_{J}\boldsymbol{T}^{J})\vvvert_{\mathcal{C}}\leq\max_{J\in\mathbb{N}^{m}}\vvvert a_{J}\boldsymbol{c}^{J}\vvvert_{\mathcal{C}}\leq C\max_{J\in\mathbb{N}^{m}}\vvvert a_{J}\vvvert_{\mathcal{A}}\cdot\boldsymbol{r}^{J}$$

By Corollary 2.5, $\gamma$ is admissible, the norm $\vvvert\mathord{\cdot}\vvvert_{\mathcal{C}}$ is equivalent to the quotient norm of the $\mathcal{A}$-Tate norm. Hence $\mathcal{C}$ is an affinoid algebra. The strictness is obtained by choosing $r_{i}\in\lvert k^{\times}\rvert$ (see Lemma 2.47). ∎

Let $\{a_{i}\}_{i\in\{1,\dots,m\}}$ be a set of generators of $A$. For each $i\in\{1,\dots,m\}$, let $r_{i}$ denote $\vvvert a_{i}\vvvert_{\mathcal{B}}$ and let $\boldsymbol{r}$ denote the multi-radius consisting of $\{r_{i}\}_{i\in\{1,\dots,m\}}$. Consider the Tate algebra $\mathcal{T}_{\boldsymbol{r}}$ and the homomorphism of $k$-algebras

$$k[T_{1},\dots,T_{m}]\rightarrow\mathcal{B},\quad T_{i}\mapsto a_{i}$$

By the ultra-metricity of $\vvvert\mathord{\cdot}\vvvert_{\mathcal{B}}$ and the definition of $\boldsymbol{r}$, one has

$$\forall n\in\mathbb{N},\forall J\in\mathbb{N}^{m},\forall f_{J}\in k,\vvvert\sum_{J\in\mathbb{N}^{m}}f_{J}\cdot\boldsymbol{a}^{J}\vvvert_{\mathcal{B}}\leq\vvvert\sum_{J\in\mathbb{N}^{m}}f_{J}\cdot\boldsymbol{a}^{J}\vvvert_{\mathcal{T}_{\boldsymbol{r}}}$$

so by a density argument one can extend it to a homomorphism of Banach $k$-algebras

$$\mathcal{T}_{\boldsymbol{r}}\rightarrow\mathcal{B},\quad T_{i}\mapsto f_{i}$$

Let $\mathscr{I}$ be the kernel ideal of this homomorphism. To conclude it suffices to take $\mathcal{A}$ as $\mathcal{T}_{\boldsymbol{r}}/\mathscr{I}$. ∎

By assumption, there exists $j\in\mathbb{N}$ and a homomorphism of $k$-algebras and elements $\{e_{i}\}_{i\in\{1,\dots,j\}}\subseteq B$ such that

$$F:\bigoplus_{i\in\{1,\dots,j\}}A\rightarrow B,1_{i}\mapsto e_{i}$$

Moreover, $F$ is bounded

$$\vvvert\sum_{i\in\{1,\dots,j\}}a_{i}\cdot e_{i}\vvvert\leq\max_{i\in\{1,\dots,j\}}\vvvert a_{i}\cdot e_{i}\vvvert\leq\max_{i\in\{1,\dots,j\}}\vvvert e_{i}\vvvert\cdot\max_{i\in\{1,\dots,j\}}\vvvert a_{i}\vvvert$$

So $F$ extends to a homomorphism of Banach $\mathcal{A}$-modules

$$\mathcal{F}:\bigoplus_{i\in\{1,\dots,j\}}\mathcal{A}\rightarrow\mathcal{B},1_{i}\mapsto e_{i}$$

Let $\mathcal{B}^{-}$ be the image of $\mathcal{F}$, it is a Banach finite $\mathcal{A}$-module with the quotient norm $\lVert\mathord{\cdot}\rVert_{\mathcal{F}}$ induced by $\mathcal{F}$. As $\mathcal{B}^{-}$ is Banach finite over $\mathcal{A}$, it is an affinoid algebra with an affinoid algebra spectral norm $\vvvert\mathord{\cdot}\vvvert^{-}$, which is equivalent to $\lVert\mathord{\cdot}\rVert_{\mathcal{F}}$. Now on $\mathcal{B}^{-}$, $\vvvert\mathord{\cdot}\vvvert$ is bounded with respect to $\vvvert\mathord{\cdot}\vvvert^{-}$ by the continuity of $\mathcal{F}$. To show the reverse, note that $\mathcal{B}^{-}$ is dense in $\mathcal{B}$, so by Theorem 2.30 one has for any $b\in\mathcal{B}^{-}$

$$\vvvert b\vvvert^{-}=\max_{z\in\mathfrak{M}(\mathcal{B}^{-})}\lvert b(z)\rvert=\max_{z\in\mathfrak{M}(\mathcal{B})}\lvert b(z)\rvert=\vvvert b\vvvert_{\mathrm{sp}}\leq\vvvert b\vvvert$$

Therefore $\vvvert\mathord{\cdot}\vvvert$ and $\vvvert\mathord{\cdot}\vvvert^{-}$ are equivalent norms on $\mathcal{B}^{-}$, so $\mathcal{B}^{-}$ is closed in $\mathcal{B}$, hence coincides with it. ∎