prove that if $G$ is a graph with $n$ vertices and $\delta(G) \geq (n - 1) / 2$ then $G$ is $\frac{n-1}{2}$-edge-connected

Question

So I know $G$ is connected since $\delta(G)\geq(n-1)/2$, we can simply prove it by contradiction. My approach to this question is use contradiction. Suppose that $G$ is not $\frac{n - 1}{2}$- edge - connected, then let $W$ be a edge set s.t $G\setminus W$ is disconnected. Then there will be at least 2 component and the vertex $v$ in a smaller component will have a degree $\leq n/2 - 1$ if $v$ is not incident to any edge in $W$. This will contradict the assumption. However, I don't know how to continue the case when all vertex is incident to $W$.

Answer 1

Let $V’$ be the set of t the vertices in the smaller component and $|V|=k\le n/2$ vertices. Then each vertex $v\in V’$ is adjacent only to vertices of $V’$, than is degree of $v$ in the residual graph is at most $k-1$. Therefore at least $\delta(G)-(k-1)\ge (n+1)/2-k$ edges incident to $v$ belong to $W$, that is $$|W|\ge k((n+1)/2-k)\ge (n-1)/2.$$