Consider $(\Re ^{2},\left \| \cdot \right \|)$ with $\left \| x_{1}^{2}, x_{2}^{2} \right \| = \sqrt{x_{1}^{2} + x_{2}^{2}}$.
Is the set $\{(x_{1} ,x_{2})\in \Re ^{2} \mid x_{1}\neq 0 \text{ and } x_{2}=1/x_{1} \}$ complete?
Now I know for a set to be complete all cauchy sequence in the set must be convergent. I need help with the identification of the cauchy in the set. I suspect that the only cauchy sequences in the set are either constant sequences or the sequences which eventually become constant i.e. with a constant tail. If that's the case then we can show the set is complete. Am I correct?
$(1+\frac 1n , \frac 1 {1+\frac 1n})$ is a counter-example to your claim.
Let $(a_n,b_n)$ be a Cauchy sequence. Then $a_nb_n=1$ for all $n$. Also the sequence is Cauchy in $\mathbb R^{2}$ (which is complete) so $(a_n,b_n)$ converges to some point $(a,b)$ in the plane. This gives $a_n \to a$ and $b_n \to b$. Take limits in $a_nb_n=1$ to see that $ab=1$. Now check that $(a,b) $ belongs to the given set and $(a_n,b_n)$ converges to $(a,b)$. Since every Cauchy sequence in the set converges the set is complete.