Since $\mathbf{B}\left(s\right)$ is a unit vector, $\mathbf{B}·\mathbf{B}\prime \=0$, so that $\mathbf{B}\prime$ is orthogonal to B. That puts $\mathbf{B}\prime$ somewhere in the osculating plane, that is, the plane formed by T and N.

Now B is orthogonal to T, so $\mathbf{B}·\mathbf{T}\=0$. Differentiating gives $\mathbf{B}\prime ·\mathbf{T}\+\mathbf{B}·\mathbf{T}\prime \=0$. Rearranging gives

where $\mathbf{T}\prime \=\mathrm{\κ} \mathbf{N}$ is the basis for the definition of the curvature $\mathrm{\κ}$.

Thus, $\mathbf{B}\prime$, already in the osculating plane, is now orthogonal to T. Hence, it must be along N, which was to be established. Incidentally, since $\mathbf{B}\prime$ is along N, it must be that it is a  multiple of N, and that multiple is taken as a measure of the torsion by writing $\mathbf{B}\prime \= -\mathrm{\τ} \mathbf{N}$.