Prove that this sequence converge. I am obliged to prove that this sequence:
$\large {a_n=(1+\frac{1}{3})(1+\frac{1}{9})(1+\frac{1}{27})...(1+\frac{1}{3^n})}$
is convergent sequence.
I mean I was thinking about this and I know that $\large\lim_{n \to \infty} (1+\frac{1}{3^n})=1 $
From this I know that it will be probably convergent sequence but I know that it is not well written proof, and probably does not prove anything. I would be glad for any tips how to prove this. 

The fact that $\lim_{n\to\infty}(1+\frac1{3^n})=1$ is only a necessary condition for convergence of the infinite product.
Hint: Have a look at the logarithm of the product and observe that for $x>0$ we have $0\le \ln(1+ x)\le x$.
