Suppose that {x(n)} 1=n=infinity, is a sequence of real numbers with limit x, and suppose that a≤ x(n) ≤b, all n. Prove that a≤ x ≤b.
If $\displaystyle x_n$ is convergent and $\displaystyle x_n \leq b$ then $\displaystyle x\leq b$ - where $\displaystyle x=\lim x_n$.
This is what we will prove. Say we prove that if $\displaystyle x_n \leq 0$ then $\displaystyle x\leq 0$. This is sufficient to complete the proof. Why? Because if $\displaystyle x_n \leq b$ with $\displaystyle x = \lim x_n$ then define $\displaystyle y_n = x_n - b$ so $\displaystyle y_n \leq 0$ and by above $\displaystyle y \leq 0 \implies x \leq b$ (here $\displaystyle y = \lim y_n$).
Now let $\displaystyle x_n\leq 0$. And assume, by contradiction, $\displaystyle x > 0$. Then there is $\displaystyle N$ so that $\displaystyle |x_N - x| < \frac{x}{2}$ and so $\displaystyle x_N > \frac{x}{2} > 0$ a contradiction.
Similary do it for $\displaystyle x_n \geq a \implies x \geq a$ by repeating the argument.