Proving a simple graph is a connected graph

Does any proof exist that a simple graph with $n$ vertices such that the least vertex degree is $\geq \frac{n-1}{2}$ is a connected graph? (i.e. does such a proof have a name?)

Suppose there is a subset of $m$ vertices that form a connected component. What is the max number of edges in this subgraph? (think of it as a complete graph on $m$ vertices)

What is the minimum number of edges in this subgraph? (think of it as an $((n-1)/2)$-regular graph with $m$ vertices)

Compare those two numbers.

Explicitly, the first number is $m(m-1)/2$, and the second is $m(n-1)/2$. You must have $m(n-1)/2 \leq m(m-1)/2$ (min $\leq$ max), and $m \leq n$ (it's a subgraph), so $m=n$.