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Direct proof of the connectedness of $B(n,k)$

My professor gave a direct proof to the fact that $B(n,k)$ is connected (for $n \geq 2$ and $0 \leq k \leq n-1$) without use of prime graphs. I tried to prove it myself but I found that it is more laborious. I have already tried to prove this by contradiction.
If we assume that there exists a connected graph $G$ such that $G \subseteq B(n,k)$, then, I can’t come up with a proof to the fact that $G$ is connected.
Moreover, we have the following picture to show that we can represent $B(n,k)$ by the union of two connected graphs $A(n,k)$ and $B(n,k) \setminus A(n,k)$:

I have actually made my own pictures, but I don’t know how to write them down.

A:

This is based on the solution by @Or there is nothing to prove for $k=0$ and for $k=n$ because $B(n,n)\cong K_n$.
Let $H$ be an induced subgraph of $B(n,k)$ with $|H|=k+1$. Since $B(n,k)$ is connected, there is a path between every pair of vertices of $H$. Choose $u\in V(H)$ and $v\in V(H)\setminus\{u\}$. We have that $v\in N(u)$ because $H$ is an induced subgraph of $B(n,k)$. Let $x$ be an arbitrary vertex of $H$ distinct from $u$ and $v$. We want to prove that $x\in N(u)$. Choose a path from $u$ to $x$ in $H$. That path has at least $k

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Free download of brock biologia dei microrganismi vol 1 microbiologia generale pdf, split of brock biologia dei microrganismi vol 1 microbiologia generale pdf free with a book reader and save it to disk. View and Download Brock’s Biologia. Biologia. Biologia dei microrganismi.

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