# If a sequence of independent random variables converges almost surely to a random variable, then that limit is almost surely a constant

Let $$\{X_n\}$$ be a sequence of independent random variables converging almost surely to a random variable $$X$$. Then how to show that $$X$$ is almost surely a constant ?

I think I somehow have to apply the Borel-Cantelli lemma for independent events, but I don't know how.

Suppose that $$X_n \to X$$ almost surely. Thus if $$\Omega_0 = \{X_n \to X\}$$ then $$\mathbb P(\Omega_0) = 1$$. Then for any fixed $$c \in \mathbb R$$ $$\Omega_0 \cap \{X < c\} = \Omega_0 \cap\{X_n < c\ \text{infinitely often}\}$$ Then apply Borel-Cantelli to justify that $$\{X_n < c\ \text{infinitely often}\}$$ happens with probability either 0 or 1, thus $$\mathbb P(X < c) = \mathbb P(\Omega_0 \cap \{X < c\}) = \mathbb P(\Omega _0\cap \{X_n < c\ \text{i.o.}\}) = \mathbb P(\text{X_n < c i.o.})$$ and will equal 0 or 1. Use this to deduce that $$X$$ is almost surely constant.