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 $\displaystyle |s_n-s|<\epsilon$. Now take $\displaystyle N_1=\text{max}\{N_0, N\}$ and $\displaystyle n>N_1$ implies

$\displaystyle |s_n-s|<\epsilon$

$\displaystyle |a-s|<\epsilon$

Since $\displaystyle x\le |x|$, we must have

$\displaystyle a-s<\epsilon$

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.