Almost stationary sequence

Question

Prove that, if $\left \{ x_{n} \right \}_{n=1}^{\infty}$ is almost stationary sequence, then $\lim_{n\to\infty}x_{n}=x$ .

Solution:

Sequence $\left \{ x_{n} \right \}_{n=1}^{\infty}$ is almost stationary if: ($\exists s \in \mathbb{N}$)($\exists x \in \mathbb{X}$) ($\forall n \geq s$) $ x_{n}=x$.

So the definition of stationary sequence tells us, that all elements of the sequence with index $\geq s$ are equal to x.

So it proves directly that $\lim_{n\to\infty}x_{n}=x$ .

Is that correct?

Answer 1

To be more exact, let $ \epsilon >0$. Then we have

$$|x_n-x|=0< \epsilon$$

for all $n \ge s.$