Big-Oh question

Question

Let $y \in \Bbb R_+$ be an arbitrary positive real number. Prove that if $f \in O(y)$, then $f \in O(1)$.

My attempt: $f \leq c_1y$, then let $c_2 = c_1y$, therefore $f \leq c_2\times 1$. Is it correct? Thanks.