= Solution
Write the increment function of the <one-step method> as $\Phi(x,v,h)=f(x+\beta h,v+\beta h f(x,v))$. Its <consistency error> is
$$
T_n=\frac{y(x_n+h)-y(x_n)}h-\Phi(x_n,y(x_n),h).
$$
As $h\to0$, the first term tends to $y'(x_n)$ and the second tends to $f(x_n,y(x_n))=y'(x_n)$ by <continuity>. Both limits are uniform along the exact-solution curve, a <compact set>: its small stage perturbations lie in a fixed compact neighborhood, so <uniform continuity> applies there. Thus $\max_{n<N}|T_n|\to0$, proving \b[consistency for every $\beta\in[0,1]$].
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