A counterexample is given by $n=2$, $C=\{(0,0)\}$, and $C_k=\{(t,kt)\colon0\le t\le1\}$ (or one can instead take $C_k=\{(t,t/k)\colon0\le t\le1\}$).
Another counterexample, in the same spirit, is given by $n=1$, $C=\emptyset$, and $C_k=\{k\}$ (or one can instead take $C_k=\{1/k\}$) -- if you don't mind $C$ being empty.