Question

Anyone good with proofs?
Show that if sn >=0 for all n and lim n => infinity sn = s then s>=0.
(using the definition of a limit).

Answer 1

Suppose \(s_n\rightarrow s\) where \(s<0\).

Then \(\forall \epsilon>0 \ \exists \ N\in \mathbb{N} \ \forall n\ge N |s_n-s|<\epsilon\)

So there is a \(N\) s.t. \(n\ge N\) implies \(|s_n-s|<\frac{|s|}{2}\implies s_n<\frac{|s|}{2}+s<0.\)

Why is this a contradiction?

Answer 2

why is sn - s < |s|/2 implied?

Answer 3

Because the sequence converges.

Answer 4

to s