Is my proof correct for this Real Analysis Problem?

Question

Prove: If s_n >= 0 for n >= k and [n-->infinity]Lim(s_n)=s, then s>=0.

My proof (by contradiction):

Suppose that s is not greater than or equal to 0. Then s does not equal 0 and s is not greater than 0; hence, s<0. But if s<0, then by choosing epsilon>0 and by letting N be a positive integer such that n>=N implies that abs(s_n-(-s))<epsilon or, equivalently,

-s - epsilon < s_n < -s + epsilon, which is not true because s_n is never less than (or equal?) to -s + epsilon; thus, since the supposed conclusion (s<0) contridicts the hypothetical fact that s_n >= 0, it must not follow from the given hypothesis.

Answer 1

It seems that you have the correct ideas, but there are some issues with the way you have written the proof. First of all, if s&lt;0, then -s&gt;0 ... really -s should not be in your proof anywhere. Second, you should choose epsilon small enough so that s + epsilon &lt; 0, which is always possible when s&lt;0. Finally, you need to take N large enough so that for n at least N |s_n - s| &lt; epsilon and n &gt;=k.