In Bourbaki's TVS Chapter IV Section 5, the last part of the proof of Proposition 2(in the 4th line from the bottom), it states that "Since $u(X)$ is a compact subset in $K$..."

Why this is true? (Actually one only needs to show the set $\{u(x_m)|m\in N \}$ is bounded for the proof to get through). Thanks in advance.