1. **Problem Statement:** Show by example that the infinite union of compact sets may not be compact. 2. **Recall the definition:** A set is compact if it is closed and bounded. 3. **Example:** Consider the sets $K_n = [0, 1 - \frac{1}{n}]$ for $n = 1, 2, 3, \ldots$. 4. Each $K_n$ is compact because it is a closed interval and bounded. 5. The infinite union is $$\bigcup_{n=1}^\infty K_n = [0, 1)$$ 6. The set $[0, 1)$ is not compact because it is not closed (it does not include the point 1). 7. **Conclusion:** The infinite union of compact sets $K_n$ is not compact, proving the statement.