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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 $g \preceq f$.

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    I tried starting with the elementary concept of showing that something was symmetric, but it came out strange, and the last part of the question is confusing me.2012-05-28
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    What is your definition of *growth function*?2012-05-28
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    Any non- decreasing function $f : [o, \infty) \rightarrow [o, \infty)$ is a growth function.2012-05-28
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    Then $\preceq$ **isn’t** symmetric; I’ll post an answer in a minute.2012-05-28
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    Okay, so I wasn't going crazy. I was getting a weird answer for the symmetric part.Thanks for your help. I will keep working on it.2012-05-28

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