Understanding CLIQUE structure

Question

I am working on the following problem:

Recall the definition of a complete graph Kn is a graph with n vertices such that every vertex is connected to every other vertex. Recall also that a clique is a complete subset of some graph. The graph coloring problem consists of assigning a color to each of the vertices of a graph such that adjacent vertices have different colors and the total number of colors used is minimized. We define the chromatic number of a graph G to be this minimum number of colors required to color graph G. Prove that the chromatic number of a graph G is no less than the size of a maximal clique of G.

So far, I have been thinking about the problem and came up with the following:

• In a clique, every vertex is adjacent to every other vertex
• Therefore, since the problem states adjacent verteces always have different colors, the number of colors needed would be the number of verteces in the clique
• Thus, the chromatic number of any graph G has to be at least the maximal size of the clique, per the definition of a clique.

Can someone verify if my understanding is correct? Or am I glaringly not understanding a clique?