Problem about Random walk and Stopping time. Here is an example in "Probability with Martingales"





My questions are:
(1)Does equation (a) hold for $T=\infty$?
(2)The equation:$$\mathbb{E}M_T^\theta=1=\mathbb{E}[(sech \theta)^Te ^\theta]$$
The author said when $T=\infty$ ,$\mathbb{E}[(sech \theta)^Te ^\theta]=0$
So the equation doesn't hold??
(3)Why if $T=\infty$ ,$(sech\theta)^T \uparrow 0$?
In my opinion,if $T=\infty$ ,$(sech\theta)^T \equiv 0$.

Thanks for regards.

The author did not say that the expectation is $0$: he rather meant that 
$M_{T(\omega)}^\theta=0$ if $T(\omega)=\infty$. This is justified because $0\lt \mathrm{sech}(x)\lt 1 $ for each $x$.
We thus have 
$$\mathbb E[M_{T}^\theta]=\mathbb E[M_T^\theta\chi\{T<\infty\}].$$
