I know that the solution is really simple but still i can't find it. Can you give me a hint first and the solution after?
2026-04-06 02:40:14.1775443214
Prove that if $\lim_{n\to\infty} a_n=a>0 $ then $a_n>0$ eventually.
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If $\lim_{n\to\infty} a_n=a>0 $ then $$\forall \epsilon>0\quad \exists M\quad \forall n>M \quad |a_n-a|<\epsilon$$ take $\epsilon={a}$ therefore $$\exists M\quad \forall n>M \quad |a_n-a|<\epsilon=a\to -a<a_n-a<a\to \quad 0<a_n<2a$$ but this doesn't mean at all that all the terms of $a_n$ are positive (take $a_n=(1+\frac{1}{n})^n-2.5$)