= Solution
Write $p_m=\mathbb P(S_T=m)$, $p_{m+1}=\mathbb P(S_T=m+1)$, and $q_m=p_m+p_{m+1}$. Part (b) gives
$$
1=2^{-m}(1-q_m)+2^mp_m+2^{m+1}p_{m+1},
$$
so $q_m\leq2^{-m}$. The exit-state decomposition also gives
$$
\frac{\mathbb ES_T}{m}=-1+2q_m+\frac{p_{m+1}}m\longrightarrow-1.
$$
Combining this with part (c), including the upper overshoot, yields
$$
\boxed{\frac{\mathbb ET_m}{m}\longrightarrow\frac74.}
$$
For example, the same calculation supplies the quantitative bound
$$
0\leq\frac74-\frac{\mathbb ET_m}{m}\leq\frac74\left(2+\frac1m\right)2^{-m}.
$$
The asymptotic linear growth is determined by the negative mean increment; the exponentially unlikely upper exit gives a vanishing correction.
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