Thread: sequence problem!

(a) Suppose that {an} is a convergent sequence of points all in [0, 1] (ie.
an∈[0, 1] for all n). Prove that limit of an as n-->∞ is also in [0, 1].

(b) Find a convergent sequence {an} of points all in (0, 1) such that limit of an as n-->∞
is not in (0, 1).

1) Let $\displaystyle \{ a_n \}$ $\displaystyle \subset [0,1]$ such that $\displaystyle a_n \rightarrow a \in \mathbb{R}$ then for all $\displaystyle \epsilon > 0$ there exists an $\displaystyle N$ such that if $\displaystyle n>N \vert a_n -a \vert < \epsilon$ then let $\displaystyle \vert a \vert \leq \vert a-a_n \vert + \vert a_n \vert < \epsilon + \vert a_n \vert \leq \epsilon + 1$ and so $\displaystyle \vert a \vert < 1 + \epsilon$ for all $\displaystyle \epsilon >0$, and so $\displaystyle \vert a \vert \leq 1$. In a similar fashion you prove that $\displaystyle a$ can't be negative.

2) Let $\displaystyle a_n =\frac{1}{n}$