= Solution
The <homogeneous treatment effect> and consistency imply
$$
Y-\beta_0A=Y(0)=:R.
$$
Instrument validity gives $Z\mathrel\perp R\mid X$.
If Assumption 1 holds, put $V=R-\alpha_0X$. Then $\mathbb E[V\mid X]=0$, so conditional instrument independence gives
$$
\mathbb E[VZ]=0,
\qquad
\mathbb E[VX]=0.
$$
Consequently both population estimating equations vanish at $(\beta_0,\alpha_0)$ for any probability limit of $\widehat\gamma$.
If Assumption 2 holds, choose the linear-projection coefficient
$$
\alpha_*=\frac{\mathbb E[RX]}{\mathbb E[X^2]}.
$$
Then $\mathbb E[(R-\alpha_*X)X]=0$, while
$$
\begin{aligned}
\mathbb E[(R-\alpha_*X)(Z-cX)]
&=\mathbb E[(R-\alpha_*X)\{Z-\mathbb E[Z\mid X]\}]\\
&\quad+(\gamma_0-c)\mathbb E[(R-\alpha_*X)X]=0.
\end{aligned}
$$
The first term is zero by conditional instrument independence and the second by the definition of $\alpha_*$. Thus $(\beta_0,\alpha_*)$ solves the population equations. Under the nonsingularity condition from part b, the root is unique, so standard <estimating equation> consistency proves
$$
\widehat\beta\xrightarrow{p}\beta_0
$$
whenever either Assumption 1 or Assumption 2 holds. This is <double robustness>.
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