Consider the random walk $\{X_k\}_{k\geq0}$ on $\mathbb{Z}$ with transition probabilities $$\begin{cases} p_{i,i-1} = p_{-1} &> 0 \\ p_{i,i+1} = p_{1} &> 0 \\ p_{i,i+2} = p_{2} &> 0 \\ \end{cases}$$ and $$\begin{cases}+p_{-1} + p_1 + p_2 &= 1 \\ -p_{-1}+p_1+2p_2 &\geq 0 \\ \end{cases}$$ while $T_n := \min\{k; X_k \geq n\}$.
(A) Show the existence of limits $$\lim_{n\to \infty} P_0(X_{T_n}=n)$$ and $$\lim_{n\to \infty} \frac{P_0(X_{T_n}=n+1)}{P_0(X_{T_n}=n)}$$
(B) Show that $$\lim_{n\to \infty} \frac{P_0(X_{T_n}=n+1)}{P_0(X_{T_n}=n)} = 1 - P_0(X_{T_1}=1)$$ and
(C) determine $$P_0(X_{T_1}=1)$$ and $$P_0(X_{T_2}=2)$$
Any help is really appreciated!