Let our ambient space be a Hilbert space $H$ over the field $\mathbb{C}$. I am reading a passage of a book on spectral theorem and I don't quite follow the given proof for the following claim:
Claim: If $A$ is essentially self-adjoint, then $A^\text{cl}$ is the unique self-adjoint extension of $A$
where $A^{\text{cl}}$ is an operator such that $\overline{G(A)} = G(A^\text{cl})$ for the graph of $A$. Consequently $A^{\text{cl}}$ is the smallest closed extension of $A$ in the sense that if $B$ is any other closed extension of $A$, then $G(A^\text{cl})\subset G(B)$.
The given proof is as follows:
Suppose $B$ if a self-adjoint extension of $A$. Since $B$ is closed, it is an extension of $A^\text{cl}$. As $B = B^*$, it follows from the definition of the adjoint that $D(B^*)\subset D(A^\text{cl})$. Thus we have $D(B^*)\subset D(A^\text{cl})\subset D(B)$. $\square$
What I don't understand is that why it follows from the definition of adjoint that $D(B^*)\subset D(A^\text{cl})$? By definition $v\in D(B^*)$ if and only if the mapping $u\mapsto \left<Bu, v\right>$ is bounded for all $u\in D(B)$, and then we can write $\left<Bu, v\right> = \left<u, B^*v\right>$. On the other hand, $v\in D(A^\text{cl})$ if and only if $\exists (u_n)_n\subset D(A)$ such that $\lim_{n\to\infty}u_n = v$ and $\lim_{n\to\infty}Au_n = w \in H$ and we define $A^\text{cl}v := w$. So why does the "definition of the adjoint" give us what we want?