# Thread:

1. ## real analysis test question

Suppose that lim s(sub n) =s, with s >0. Prove that there exists an N in R (real numbers) such that s(sub n) >0 for all n in N.

2. Hello,
Originally Posted by trojanlaxx223
Suppose that lim s(sub n) =s, with s >0. Prove that there exists an N in R (real numbers) such that s(sub n) >0 for all n in N.
$\lim s_n=s$ means that :
$\forall \epsilon>0, \exists N \in \mathbb{N}, \forall n\geq N, |s_n-s|< \epsilon$

Now take $\epsilon$ such that $s-\epsilon$ is positive. You can always find one, such as $\epsilon=\frac s2$

From the inequality $|s_n-s|<\epsilon$, we can say that $s-\epsilon<s_n<s+\epsilon$
Since we defined $\epsilon$ such that $s-\epsilon>0$, it's easy to conclude.
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