Let G be a graph with n vertices where every vertex has degree at least n/2. Prove that G is connected. (Note: Do not use the result on Hamilton cycles.)
Pick 2 vertices $a$ and $b$. We must find a path between them. If there is an edge between $a$ and $b$, then we are done. Otherwise, all of the neighbors of $a$, and all of the neighbors of $b$, must lie among the other $n-2$ vertices. $a$ and $b$ must connect to at least $n/2$ of these vertices, and can not connect to at most $n/2 - 2$. Since $n/2>n/2-2$, there must be some vertex $c$ that is connected to both $a$ and $b$, and the graph must be connected.
Consider any two complementary subsets $A$ and $V\setminus A$ of the vertices, with $a$ and $n-a$ points. One of them is $\le n/2$, wlog. we can assume it is $a$.
If $A$ was a disjoint subgraph, then its vertices would have degree at most $a-1$ which is $<n/2$.