# Application of Strong Law of Large Numbers

I have troubles with the following problem. Can you help me, please?

Suppose $$\lambda_1 = 2$$ and $$\lambda_2 = \frac{1}{3}$$ and $$\lambda_1$$, $$\lambda_2$$ are chosen independently with probability $$1/2$$ each and multiply the number 1 i.e. we have a random sequence $$\lambda_{i_1}1, \lambda_{i_2}\lambda_{i_1}1, \dots, \lambda_{i_n}\dots\lambda_{i_1}1$$ where each $$\lambda_{i_k} \in \{ \lambda_{1}, \lambda_{1} \}$$.

How to show that with probability $$1$$, $$\lambda_{i_n}\dots\lambda_{i_1}1 \rightarrow 0$$?

You could use a logarithm to turn the product into a sum. It will prove, that $$P(\lim_{n\to\infty}\sqrt[n]{1\lambda_{i_1}...\lambda_{i_n}} = \tfrac{2}{3}) = 1$$ if i'm not mistaking...