$\Delta$ stands for the maximum degree in a graph and $\delta$ stands for the minimum degree in a graph.
Hint: First show that there must be some component that contains at least $\Delta + 1$ vertices. Then show that if there were any other component, these two components together would contain more than $n$ vertices.)
Using the hint I tried this below:
Let $G$ be a graph with maximum degree $\Delta$. Assume $G$ is disconnected. Suppose every component of $G$ has at most $\Delta$ vertices. Then every component has maximum degree $\le \Delta - 1.$ But $G$ must have maximum degree $ \Delta.$ Contradiction. Now that we know there exist at least one component with $\Delta + 1 = n$ vertices we conclude we can't have have any more components because even the smallest component adds one vertex to the component with $n$ vertices.
I am having trouble fitting the hypothesis $(\delta + \Delta \ge n - 1)$ into the proof. I think $\Delta + 1 = n \implies \delta + \Delta \ge n - 1,$ but the converse is not true.
Can someone please help me complete this proof. Thanks.