Comment by lzralbu

Note that $A_k = \left[\frac{1}{k}, \frac{2}{k}\right] \times \left[\frac{1}{k}, \frac{2}{k}\right] \to \{(0, 0)\}$ in $[0,1]\times [0, 1]$ with the Hausdoff metric, so it is a Cauchy sequence in $X = (0,1) \times (0, 1)$ but without convergent subsequence, i.e., $F(X)$ is not complete, hence not compact so the finite subcover argument fails.