Equivalent condition of a convex function over a vector space

Question

Let $f$ be a real-valued function on a convex subset $C$ of a real or complex vector space $E$. Show that the following are equivalent.

1. $f$ is a convex function on $C$.
2. The set $\{(c,t)\in C \times R : f(c)<t\}$ is a convex subset of $E \times R$.

I don't know how to show that 2 implies 1. Also, I would like to know how to understand this intuitively(say geometrically).