Limits Analysis: {a_n} <= 0; if {a_n}->A, then A <= 0

[Jenny4]

Let $\displaystyle (a_n)$ be a sequence where for all natural numbers n, $\displaystyle (a_n) \leq 0$. Prove that if $\displaystyle (a_n)$ is convergent with limit A, then $\displaystyle A \leq 0$.

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It sounds obvious, but I don't know how to do it. I tried doing a proof by contradiction but it went horribly wrong.

Please help x

 

[pka]

Re: Limits Analysis

Suppose that A>0. In the definition of convergence use A/2 as epsilon.
Expand the absolute value and you will see a contradiction.

 

[Jenny4]

Re: Limits Analysis

Suppose that A>0. In the definition of convergence use A/2 as epsilon.
Expand the absolute value and you will see a contradiction.

Excellent, thank you very much. x