Exercise 2.7.3

Suppose $a_n,b_n\ge 0$, $a_n\le b_n$ and define $s_n=a_1+\cdots +a_n$, $t_n=b_1+\cdots +b_n$.

(a)

We have $|a_{m+1}+\cdots +a_n|\le |b_{m+1}+\cdots +b_n|<𝜖$ implying ${\sum <!--\; FUNCTION\; APPLICATION\; -->}_{n=1}^{\infty }{a}_n$ converges by the Cauchy criterion. The other direction is analogous, if $(s_n)$ diverges then $(t_n)$ must also diverge since $s_n\le t_n$.

(b)

Since $(t_n)\to t$. This implies that $s_n$ is bounded, and since $s_n\le t_n$ implies $s_n\le t$ by the order limit theorem, we can use the monotone convergence theorem to conclude $(s_n)$ converges.