Compact subsets of infinite projective space

Question

Let's define infinite projective space space $\mathbb{CP}^{\infty}$ as direct limit $\lim \limits_{\rightarrow} \mathbb{CP}^n$. In a class I attended it was claimed that every continuous map $S^k \to \mathbb{CP}^{\infty}$ is actually valued in some subspace $\mathbb{CP}^N$ for some sufficiently high $N$. How to prove this?

Answer 1

Here's a very general statement which is enough for your situation:

Let $X_i \to X_{i+1}$ be a sequence of embeddings of $T_1$ spaces. Then any compact subset $K\subset \mathrm{colim}_i X_i$ is actually contained in an $X_i$.

A proof can be found here : Compact subset in colimit of spaces

It's a very general statement, and the proof is not hard to follow.