MHB Show that a quantified statement is true:

[zethieo]

for example this question:
∃λ∈R+, ∃m∈Z+,∀n∈m..+∞,2n+100≤λn

To be honest I'm really struggling to understand this math, and I'm actually not even totally sure what this question is called. If anyone could explain this to me or point me to some good tutorials I'd really appreciate it.

 

[Evgeny.Makarov]

This seems like the statement saying that $2n+100$ is $O(n)$. If you don't know what the big-O notation is, please ignore this.

The statement says that even though $2n+100>n$ for all $n>0$, we can find a positive constant $\lambda$ such that $2n+100\le\lambda n$. For example, $\lambda=3$ looks promising. The second subtlety is that $2n+100\le3n$ does not hold for all $n>0$, but only eventually, i.e., from some point $m$ on. Can you find such $m$ if $\lambda=3$?

 

[zethieo]

So that would mean m = 100, which proves the statement to be true.

This doesn't seem that bad, thanks for the help.