= Solution
<Continuity> and nonvanishing on the <compact> <interval> imply
$$
0<m:=\min_{[0,1]}|\alpha|\le\max_{[0,1]}|\alpha|=:M<\infty.
$$
Hence $m\|f-g\|_\infty\le d_\alpha(f,g)\le M\|f-g\|_\infty$. A $d_\alpha$-<Cauchy sequence> is therefore uniformly <Cauchy sequence>[Cauchy], so (i) gives a <continuous> <uniform limit>; the upper bound then gives convergence in $d_\alpha$. Thus $\boxed{(C([0,1]),d_\alpha)\text{ is complete}}$. The positive lower bound is what makes the argument work.
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