
Just a matter of rigor
My teacher really takes account about the rigor of a proof. The following is the proposition and a proof given by myself. Can someone please check it whether it is rigorous enough?
Proposition: Let $\displaystyle (s_n)$ be a sequence that converges to $\displaystyle s\in \mathbb{R}$. Show that if $\displaystyle s_n\ge a$ for all $\displaystyle n>N$, then $\displaystyle s\ge a$.
Proof: Let $\displaystyle \epsilon>0$, then there exists $\displaystyle N_0$ such that $\displaystyle n>N_0$ implies $\displaystyle s_ns<\epsilon$. Now take $\displaystyle N_1=\text{max}\{N_0, N\}$ and $\displaystyle n>N_1$ implies
$\displaystyle s_ns<\epsilon$
$\displaystyle as<\epsilon$
Since $\displaystyle x\le x$, we must have
$\displaystyle as<\epsilon$
Since the difference can be taken arbitrarily small, or even negative. This proves that $\displaystyle s\ge a$.
I really doubt the rigor of this proof. And after I type in the proof, I doubt the validity.(Doh)

Here is an easy proof.
If it were true that $\displaystyle s<a$ then $\displaystyle \varepsilon = as$.
From that we get $\displaystyle \left {s_Ns} \right < \varepsilon \, \Rightarrow \,s_N<a$.
What is wrong to that?