One not so simple question of probability

The sum of i.i.d random variable is one of most classical topic in probability. However, there are always questions not so easy even in the first year course.

Question : Let $X_n$ be a series of independent random variables and $S_n = \sum_{i=1}^{n}X_i$. We would like to study the behavior of $\limsup S_n, \liminf S_n$ in the following two situations : 

(1)$$\begin{eqnarray}\mathbb{P}[X_n = n] &=& \frac{1}{n+1} \\ \mathbb{P}[X_n = -1] &=& \frac{n}{n+1}\end{eqnarray}$$ 

(2)$$\begin{eqnarray}\mathbb{P}[X_n = n^2] &=& \frac{1}{n^2+1} \\ \mathbb{P}[X_n = -1] &=& \frac{n^2}{n^2+1}\end{eqnarray}$$ 

In another word, we would like to study the behavior of a random walk.

Even we have some advanced tools, we may make mistakes in this questions. Here, I give two false proofs that I have thought.

False proof (1) : OK, the sum $S_n$ is a martingale, so we apply the representation of martingale and we know that we can embed this martingale to the Brownian motion by Dubins-Schwartz theorem. (It's false since it's discrete and we may choose some specific moment in the random process)

False proof (2) : We use the theorem of optional stopping theorem to the stopping time $T = T_a \wedge T_{-b}$ and study it as the ruin of gambler. (This time the errors in the proof is harder to find. In fact, since the positive jump becomes bigger and bigger, even the martingale $S_{n \wedge T}$ isn't U.I. This isn't the same case as simple random walk or continuous martingale) 

Let's give a correct proof :

The question (2) isn't so difficult. We apply Borel-Cantelli lemma that

$$\sum_{n=1}^{\infty} \mathbb{P}[X_n = n] = \sum_{n=1}^{\infty} \frac{1}{n^2+1} < \infty$$

this means that the positive only appears finite times almost surely. So, we have of course $\lim_{n \rightarrow \infty} S_n = - \infty$.

The question (1) has an easy part. We apply once again the Borel-Cantelli lemma that 

$$\sum_{n=1}^{\infty} \mathbb{P}[X_n = n] = \sum_{n=1}^{\infty} \frac{1}{n^2+1} = \infty$$

so it will happen infinite times. Moreover, we know once the positive jump occurs, it will cover all the negative jump before this step. So as the positive jump accumulates, we conclude that

$$\limsup_{n \rightarrow \infty} S_n = \infty$$

The hard part, we apply another sub-martingale $\exp(- \lambda S_{n \wedge T})$. This time, a very big positive jump doesn't matter as we know this guy is smaller than $\exp(\lambda b)$. So by the optional stopping theorem, we get

$$(1 - \mathbb{P}[T_b < T_a])e^{-\lambda a} + \mathbb{P}[T_b < T_a] \geq 1 \Rightarrow \mathbb{P}[T_b < T_a] \geq \frac{1- e^{-\lambda a}}{e^{\lambda b} - e^{- \lambda a}}$$

We take $a \rightarrow \infty$ and get $\mathbb{P}[T_b < \infty]  \geq e^{-\lambda b}$ then we take $\lambda \rightarrow 0$ and get $\mathbb{P}[T_b < \infty] = 1$.