Assume $c_{0, \lambda}$ is a metric space of converging to zero sequences, where
$c_{0, \lambda} =\{ x = (x_i):\lim_{i \rightarrow \infty} i^{\lambda} x_i = 0 \}$,
here $\lambda > 0$ imposes a certain restriction on the convergence speed for the sequences; with the following metric
$d(x,y) = \sup_{i}i^{\lambda}|x_i - y_i|$,
I'm trying to prove that such space is complete.
My approach:
Assume that $(x_n)$ is a Cauchy sequence from $c_{0, \lambda}$. It is clear that $(x_n)$ is also from $c_{0}$ (with $\lambda = 0$, which is known to be complete). If we construct a sequence $z_n = (z_{n,i}) = ( i^{\lambda} x_{n, i} )$, by definition it should hold that $z_n \in c_0$ for some fixed $\lambda > 0$, and it's also a Cauchy sequence in $c_0$. Hence, there exists a limit $z_0 \in c_0$, which implies that $\lim_{i \rightarrow \infty} z_{0, i}= 0$. Therefore, it should hold that $\lim_{i \rightarrow \infty} z_{0,i}/{i^{\lambda}} = 0$ for the same fixed $\lambda > 0$, suggesting that $x_0 := (x_i) = (z_{0,i}/i^{\lambda}) \in c_{0, \lambda}$.
Since
$d(x_n, x_0) = \sup_{i}i^{\lambda}|x_{n,i} - x_{0,i}| = sup_i | i^\lambda x_{n,i} - i^{\lambda} x_{0, i}| = sup_{i} | z_{n, i} - z_{0, i} | \rightarrow 0$, $i \rightarrow \infty$,
it seems that $x_n \rightarrow x_0, n \rightarrow \infty$, suggesting that $c_{0, \lambda}$ is complete. Here the last result holds due to convergence in $c_0$ under supremum metric.
My question:
Is such approach suitable for proving completeness? If not, what would be the right way to proceed in this case (and in more general ones, where the restriction $i^{\lambda}$ may have various other expressions (and hence the metric would be constructed accordingly))?
The goal I'm trying to achieve here is not only to just prove the completeness for this particular example, but to better understand proving completeness for various spaces, where it is clear that it is a some sort of a restricted version of a larger, well known space the properties of which is known.
Would be grateful for any hints or corrections!