1. Limit Question

How to i prove this?

If Xn -> X and X > 0, prove that there exists a natural number K such that
X/2 < Xn < 2X

for all n in K. (In particular, Xn > 0 for all n in K.)

2. Originally Posted by guess
How to i prove this?

If Xn -> X and X > 0, prove that there exists a natural number K such that
X/2 < Xn < 2X

for all n in K. (In particular, Xn > 0 for all n in K.)
You observe that for $\displaystyle n$ large enough that

$\displaystyle |X_n-X|<X/2$

CB

3. The definition of "limit of a sequence" is "$\displaystyle X$ is the limit of the sequence $\displaystyle \{X_n\}$ if and only if for any $\displaystyle ]epsilon> 0$, there exist N such that if n> N, then $\displaystyle |X_n- X|< \epsilon$.

If X> 0 then so is X/2. Apply the definition of "limit of a sequence" with $\displaystyle \epsilon= X/2$.

If $\displaystyle |X_n- X|< X/2$ then $\displaystyle -X/2< X_n- X< X/2$ so that $\displaystyle X/2< X_n< 3X/2< 2X$.