Another Gambler's Ruin Problem

Let $(X_n)_n$ be a sequence of i.i.d. random variables with $$\mathbb{P}(X_1 = 1) = \mathbb{P}(X_1 = -1) = 1/2.$$ Define $S_n = X_1 + \cdots + X_n$ for $n \geq 1$ and $S_0 = 0$.We can easily show that $$\mathbb{P}(S_n = N \text{ before } S_n = -1) = \frac{1}{N+1}.$$ The strategy is defining the function $f:\{-1, ..., N\} \to [0, 1]$ by the rule $$f(a) = \mathbb{P}(S_n+a = N \text{ before } S_n+a=-1).$$ One can show that $$f(a) = \frac12f(a-1) + \frac12 f(a+1),$$ and thus $$f(a) = \frac{a+1}{N+1}.$$ I would like to show that $$\mathbb{P}\left(\bigcup_{n=1}^\infty \{S_n = -1\}\right) = 1.$$ Can I do this using my work above? The only luck I have is using the fact that $1 = f(a-1) + f(N-a)$.