gambler's ruin expected stopping time

Question

Let $(Y_n)_{n \in \mathbb{N}^*}$ be a sequence of independent and identically distributed random variables following Bernoulli distribustion with parameter $p$. Let $(a,b) \in (\mathbb{N}^*)^2$. For all $n\in \mathbb{N}^*$ we put $$ S_n = a + \sum_{i=1}^n Y_i $$ and $$ T = \inf \lbrace{ n \in \mathbb{N}^* \big\vert S_n \in \lbrace 0,a+b \rbrace \rbrace } $$ Let $\mathbb{E}_a(T) = \mathbb{E}(T \vert S_0=a)$. Supposing that $ \mathbb{E}_a(T) $ exists for all $a$, we can find its value using the relation: $$ \mathbb{E}_a(T)= 1+p\mathbb{E}_{a+1}(T) + (1-p) \mathbb{E}_{a-1}(T) \quad (*) $$ $$ \mathbb{E}_0(T) = \mathbb{E}_{a+b}(T)=0 $$ My question: Since $(*)$ is satisfied even if $\mathbb{E}_a(T) = +\infty$. How can I prove that $\mathbb{E}_a(T)< +\infty$ for all $a$ without using martingales. I tried with this: $$ \mathbb{E}_a(T) = \sum_{n=1}^{+\infty} \mathbb{P}\lbrace T>n \rbrace $$ but i couldn't achieve the result.