$D:=\{(x,y):x^2+y^2<1\}$ is complete? How to conclude whether the set
$D:=\{(x,y):x^2+y^2<1\}$ is complete ?
I thought in the straight forward process of using a Cauchy sequence say $(x_n,y_n)$ then could not proceed further?

The space is not complete. Any Cauchy sequence in the disk that converges to a point on the boundary will provide a counterexample.
For example, consider $a_n = (1-1/n, 0)$. This sequence is Cauchy, since if $\epsilon>0$. we have $d(a_m,a_n)<\epsilon$ whenever $m,n>1/\epsilon$. The sequence converges to $(1,0)$, which is outside $D$, so $D$ is not complete.
