Let $a,b\in B_X$ then $\|a\|\leq1,\|b\|\leq1$. Since for all $0\leq\lambda\leq1,$ \begin{align*} \|(1-\lambda)a+\lambda b\|&\leq\|(1-\lambda)a\|+\|\lambda b\|\\ &=(1-\lambda)\|a\|+\lambda\|b\|\\ &\leq1-\lambda+\lambda\\ &=1 \end{align*} then $(1-\lambda)a+\lambda b\in B_X$.\\ Since $\|-a\|=\|(-1)a\|=|(-1)|\|a\|=\|a\|\leq1$ then $-a\in B_X$.\\ Therefore, $B_X$ is both symmetric and convex.