# Thread: how to show xn>0 need not imply lin xn>a

1. ## how to show xn>0 need not imply lin xn>a

Let (x_n) be a convergent sequence, and let a,b $\displaystyle \in$ R.

Show that for all n, x_n>a ned not imply limx_n>a.

How do I show this when I have previously proven that If x_n>= a for all n, then lim x_n>=a. (this was another question that i had proven before hand) But how would I show it, when it contradicts the previously stated question.

2. What about the sequence $\displaystyle \displaystyle \left\{\frac{1}{n}\right\}$. This is $\displaystyle \displaystyle > -1 \forall n \in \mathbf{Z}^+$ but $\displaystyle \displaystyle \to 0$ as $\displaystyle \displaystyle n \to \infty$

3. The demonstration that $\displaystyle \displaystyle \lim_{n \rightarrow \infty} x_{n} = a$ doesn't imply in any case that $\displaystyle \forall n$ is $\displaystyle x_{n}>a$ or $\displaystyle x_{n}<a$ [the convergence may be 'oscillating'...] but that is $\displaystyle \lim_{n \rightarrow \infty} |x_{n}-a|=0$...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$