real analysis - Showing that a sequence converges in norm

I am trying to prove the following:

Let $X$ be a normed linear space satisfying the property: $\forall \left\{x_n\right\}, \left\{y_n\right\} \subseteq X $, we have
$\|x_n\|=\|y_n\|=1, \|x_n+y_n\|\rightarrow 2 \Rightarrow \|x_n-y_n\|\rightarrow 0.$

If $\left\{z_n\right\} \subseteq X$ converges to $z\in X$ weakly (meaning $\displaystyle \lim_{n\rightarrow \infty} f(z_n)=z$ for all $f\in X^*$) and $\|z_n\| \rightarrow \|z\|$, then $\|z_n-z\|\rightarrow 0$.

Here is what I am trying to do:

I can consider $\left\{z \right\}$ as a sequence in $X$. I want to show that $\|z_n+z\|\rightarrow 2$. Well, since $\|z_n\| \rightarrow \|z\|=1$, then since $\|z_n+z\|\leq\|z_n\|+\|z\|$, then $\displaystyle \lim_{n\rightarrow \infty} \|z_n+z\| \leq 2\|z\|=2$.

I can't figure out how to possibly show that $\displaystyle \lim_{n\rightarrow \infty} \|z_n+z\| \geq 2$. How would I even incorporate the weak convergence assumption? Any help would be greatly appreciated! Thank you.