Converging Sequence' Unique Limits

Theorem

• If $\{a_n\}$ converges
• Then, $\{a_n\}$‘s limit is unique

Proof

1. Suppose $\{a_n\}$ converges
2. Choose arbitrary $l_1,l_2$ s.t $\{a_n\}=l_1$ and $\{a_n\}=l_2$
3. We want to show that $l_1=l_2$
   1. $⟹l_1-l_2=0$
   2. $⟹\forall ϵ>0,|l_1-l_2|<ϵ$
4. Let $ϵ>0$ be arbitrary
5. Invoke 1 with ${ϵ}_1=\frac{ϵ}{2}$
6. Note that $\exists N>0$ s.t if $n>N$ then $|a_n-l_1|<{ϵ}_1$
7. Invoke 1 with ${ϵ}_2=\frac{ϵ}{2}$
8. Note that $\exists N>0$ s.t if $n>N$ then $|a_n-l_2|<{ϵ}_2$
9. Then, consider $|l_1-l_2|=|-a_n-l_1+a_n-l_n|\le |a_n-l_1|+|a_n-l_2|=\frac{ϵ}{2}+\frac{ϵ}{2}=ϵ$
10. $\square$