I want to look at compactification of $\mathbb{C}^{n}$. There is a hypothesis that it is a complex projective space: $$\mathbb{CP}^n = \mathbb{C}^n + \mathbb{CP}^{n-1}_{\infty}$$ where $\mathbb{CP}^{n-1}_{\infty}$ are some "infinity" points. For example, for $\mathbb{C}$ the compactification is well known Riemann Sphere: $$\mathbb{CP}^1 = \mathbb{C} + \lbrace{\infty\rbrace}$$ Firstly, I need to describe $\mathbb{CP}^{n-1}_{\infty}$. Then, I want to introduce a metric on this space. By analogy, I can build a sphere $S^{2n+1}$ and consider complex lines, intersecting sphere, and then introduce a metric as the distance between the circles generated by this intersection. I find the Fubini-Study metric. Why do I need a metric? I want to show that what came out is really a compact. How can I prove that $\mathbb{CP}^n$ is compact?
Thank you for advance for your help and explanations!