Let $X$ be a topological vector space and $p$ be a semi-norm on $X.$ Show that the following statements are equivalent.

$(1)$ $p$ is continuous.

$(2)$ $p$ is continuous at $0.$

$(3)$ The set ${x in X | p(x) lt 1 }$ is open in $X.$

How do I prove $(2) implies (3)$ and $(3) implies (1) $? If we can show that $(2) implies (1)$ then we are done with $(2) implies (3).$ Because the set in $(3)$ is just $p^{-1} (B(0,1)).$