1. ## limit proof

I am really have trouble with these proofs involving limits, and would really appreciate some help. Here is an example of a problem I am having trouble with:
$
\mbox{Prove that if lim}(x_n)=x\ \mbox{and if}\ x>0\ \mbox{then there exists a natural number}\ \mbox{M such that}\ x_n>0\ \mbox{for all}\ n \geq M$

thank you!!

2. In the definition of sequence convergence use $\varepsilon = \dfrac{x}{2}>0$

Then get $-\dfrac{x}{2}.

Carry on.