Continuity of $H_s \subset C_0^k$ inclusion

Question

I've been trying to show that, if $H_s \subset C_0^k$, then the inclusion is continuous.

I don't have the Sobolev embedding theorem condition $s>k+n/2$.

$||f||_{C_0^k} =\sum_{|\alpha|\leq k} ||\partial ^{\alpha}f||_u$.

$||f||_{H_s}$ is the s-Sobolev norm.

I tried to argue with sequences, but it's not a trivial relation between the norms.

I did find a similar question here, but it wasn't properly answered.

Any suggestions are welcome. Thanks.