sillyus sodus
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Homework Statement
Given a convergent sequence:
a_n \rightarrow a
and a continuous function:
f:\mathbb{R}\rightarrow\mathbb{R}
show that there exists an N\in\mathbb{N} such that \forall n>N:
f(a_n)\geq\frac{f(a)}{2}
Homework Equations
Usual definitions for limit of a sequence and continuous function.
The Attempt at a Solution
I've tried playing around with it but I really don't understand what to do, I know its an easy question and I feel pretty stupid but I'm stuck. Can anyone just get me started on this one?
So far I've rearranged it so it looks like:
f(a)-f(a_n)\leq f(a_n)
Which sort of looks like a limit without the abs values...
I figure that the function of a sequence is a sequence again, btu what am I trying to say here? That it is a divergent sequence?