陶哲轩实分析（上）9.10及习题-Analysis I 9.10

limits at infinity, 也是很短的一节，但其事实上起到了将极限定义从R延拓到R*的作用.

Exercise 9.10.1

First suppose $\lim_{n→+∞;n∈\mathbf N}a_n=L$, then $∀ϵ>0,∃M>0$ s.t.
$|a_n-L|<ϵ,\quad ∀n∈\mathbf N,n>M$
We let $N=[M]+1>M$, then if $n>N$, we shall have $|a_n-L|<ϵ$, this means $\lim_{n→∞}a_n=L$.
Conversely, suppose $\lim_{n→∞}a_n=L$, then $∀ϵ>0,∃N∈\mathbf N,N>0$ s.t.
$|a_n-L|<ϵ,\quad ∀n∈\mathbf N,n>N$
We choose $M=N$ in Definition 9.10.3, then we can see $\lim_{n→+∞;n∈\mathbf N}a_n=L$.