Complete metric space question

Question

I am stuck on this question. I think that the answer for both (i) and (ii) should be the same, since alpha is continuous on $[0,1]$, so bounded.

Answer 1

They are both complete. (i) is a standard exercise, and (ii) follows from (i), since $$\inf |\alpha| ‖ f - g ‖_∞ ≤ \text{d}_\alpha (f,g) ≤ \sup |\alpha| ‖f-g‖_∞ $$ where $\sup|\alpha|,\inf|\alpha|>0$ by continuity on a compact interval. So a $\text{d}_\alpha$-Cauchy sequence $f_n$ will converge to some continuous function $f$ in $∞$ norm and hence also wrt the $\text{d}_\alpha$ metric.