Letting $\lbrace X_n \rbrace$ be a sequence of independent real valued random variables and supposing that $$\mathbb{P}\left( \lim_{n \to \infty} \frac{X_1+...+X_n}{n}=1 \right) \gt 0,$$ I'm trying to prove that $$\mathbb{P} \left(\lim_{n \to \infty} \frac{X_1+...+X_n}{n}=1 \right) = 1.$$ That makes me think about the Kolomogorov's 0-1 Law but I can't show that the event $$\lbrace \lim_{n \to \infty} \frac{X_1+...+X_n}{n}=1 \rbrace$$ is a tail event (that is that it lies in $\bigcap_{n=1}^{\infty} \sigma \lbrace X_i:i \geq n\rbrace$).
Am I on the right track? Could you help me complete this, I'm confused!
Yes, you are on the right track. The only thing you need to prove is that $A = \left[\lim_{n \to \infty} \frac{X_1 + \cdots + X_n}{n} = 1\right]$ is a tail event.
For each fixed $m$, observe that \begin{align} & \left[\lim_{n \to \infty} \frac{X_1 + \cdots + X_n}{n} = 1\right] \\ = & \left[\lim_{n \to \infty} \frac{X_1 + \cdots + X_{m - 1} + X_m + \cdots + X_n}{n} = 1\right] \\ = &\left[\lim_{n \to \infty} \frac{X_m + \cdots + X_n}{n} = 1\right] \in \sigma(X_m, X_{m + 1}, \ldots), \end{align} in view of $X_1 + \cdots + X_{m - 1}$ is a fixed random variable.