Are these metric spaces complete?

Question

Are the following metric spaces complete?

(1) $(X_f, d_f)$, where $X_f=\{0,1,2,3\}$, $d_f(x,y)=|x-y|$

(2) $(X_d, d_d)$, where $X_d=\mathbb{N}$, $d_d(x,y)=|x-y|$

Since these spaces do not seem to have any Cauchy sequences, does this mean they are vacuously complete? Alternatively, these metric spaces only have open sets, so we can't verify their completeness topologically either.

Would appreciate some clarification.

Answer 1

The only cauchy sequences in those spaces are the constant sequences (because $|x-y| \geq 1$ if $x \neq y$). Therefore they are trivially convergent and your spaces are complete