= Solution
Linear dependence means there are constants $c_1,c_2$, not both zero, with $c_1u_1+c_2u_2=0$. Since $u_1$ is nowhere zero, $c_2\ne0$, so $u_2=Cu_1$ and $u_2'=Cu_1'$. Substitution gives $W=u_1Cu_1'-Cu_1u_1'=0$.
The nonvanishing assumption is more than is needed for this direction. It also gives the converse directly: if $W=0$, then $(u_2/u_1)'=W/u_1^2=0$, so the ratio is constant on a connected interval. For arbitrary differentiable functions with zeros, a vanishing <Wronskian> alone needs more care; here the homogeneous linear equation also supplies an appropriate uniqueness theorem.
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