Commutator subgroup and a center

Hi, all

$\displaystyle C_{i}(G)$ denotes a center in central series and [G, G] denotes a commutator subgroup of G.

My textbook says $\displaystyle [C_{i}(G), C_{i}(G)] < C_{i-1}(G)$.

Since $\displaystyle C_{i-1}(G)$ is a normal subgroup of $\displaystyle C_{i}(G) $ and $\displaystyle C_{i}(G) / C_{i-1}(G) $ is abelian, it is plausible that $\displaystyle C_{i-1}(G) $ contains $\displaystyle [C_{i}(G), C_{i}(G)]$.

What bothering me is that why $\displaystyle [C_{i}(G), C_{i}(G)] $ is a proper subgroup of $\displaystyle C_{i-1}(G) $ rather than being just a subgroup?

I will appreciate if someone gives me an intuitive enlightment of the above one.