Show that a Banach space with certain assumptions must be finite dimensional

Question

I am trying to prove the following:

Let $X$ be a Banach space and $X^*$ its dual. Suppose that there is a sequence of $L_n \in X^*$ satisfying the following property: any sequence $x_j, x \in X$ converges weakly iff $L_n(x_j) \rightarrow L_n(x)$ for all $n$. Show that $X$ is finite dimensional (hint: consider $d(x, y) = \sum_{n} 2^{-n}\frac{|L_n(x-y)|}{1 + |L_n(x-y)|}$).

To prove this, I want to use the following theorem:

Any infinite dimensional Banach space endowed with the weak topology cannot be homeomorphic to a complete metric space.

So, I am trying to show that

1. $(X, d)$ is a complete metric space
2. $X$ equipped with the weak topology is homeomorphic to $X$ with the metric topology given by $d$.

which I need some helps.
I know that there is a stronger version of the theorem I mentioned above (i.e. infinite dimensional Banach space with the weak topology is not metrizable), so using this fact, it would suffices to show that 2. But, I would like to see how to prove 1 if it is indeed true.