Prove that if $G$ is a graph with $n$ vertices, where $δ (G) ≥ \frac{n - 1}{2}$ , so $G$ is connected.

Question

I would be very grateful for help with this proof:

"Prove that if G is a graph with n vertices, where $δ (G) ≥ \frac{(n - 1)} {2}$ , so $G$ is connected."

I know, that: $δ (G)$ ... is the minimum degree of the graph $G$

and I think, that maybe i can use this theorem:

If $G$ is a graph of order $n$ such that for every pair of nonadjacent vertices $u$ and $v$, $deg$ $u$ $+$ $deg$ $v$ is greater than or equal to $n-1$, then $G$ is connected

Could anyone please advise me how to prove this proof correctly, or if it can be done differently and better ?
Thank you for any advice.

Answer 1

Suppose for a contradiction that $G$ is not connected. Take a vertex $v$ on one connected component $C_{1}$. There must be at least $(n-1)/2$ additional vertices on the same component $C_{1}$ as $v$, for a total of $1 + (n-1)/2$ vertices on $C_{1}$. Now let $u$ be on some other component $C_{2}$. By a similar argument, $C_{2}$ must have at least $1 + (n-1)/2$ vertices.

Now at least how many vertices are in $C_{1} \cup C_{2}$? Derive your contradiction.