I am having a hard time trying to solve the following problem. I am very novice to the topic of growth functions at the moment. I could use a little help with this problem.
Let $\preceq$ be the relation on growth functions defined by $f \preceq g$ if there is a constant $\lambda \ge 1$ such that $f(x) \le \lambda g(\lambda x + \lambda) + \lambda$ for all $x \in [0, \infty)$. Show that $\preceq$ is symmetric and transitive and that the relation $f \sim g$ - is defined to mean both $f \preceq g$ and $g \preceq f$.