Chapter 9, Problem 9RE

Are the following statements true or false? If true, state a theorem to justify your conclusion; if false, then give a counterexample.

$\begin{array}{l}\text{(a) }\text{If }{\displaystyle \sum u_k}\text{Converges,}\text{then}{u}_k\to 0\text{as}k\to +\infty .\\ \text{(b)}\text{If}{u}_k\to 0\text{as}k\to +\infty ,\text{then}{\displaystyle \sum u_k\text{converges}\text{.}}\\ \text{(c)}\text{If}f(n)=a_n\text{for}n=1,2,3,\mathrm{...},\text{and}\text{if}{a}_n\to L\text{as}n\to +\infty ,\text{then}f(x)\to L\text{as}x\to +\infty .\\ \text{(d)}\text{If}f(n)=a_n\text{for}n=1,2,3,\mathrm{...},\text{and}\text{if}f(x)\to L\text{as}x\to +\infty ,\text{then}{a}_n\to L\text{as}n\to +\infty \\ \text{(e)}\text{If}0<a_n<1,\text{then}\left\{a_n\right\}\text{converges}\text{.}\\ \text{(f)}\text{If}0<u_k<1,\text{then}{\displaystyle \sum u_k}\text{converges}\text{.}\\ \text{(g)}\text{If}{\displaystyle \sum u_k}\text{and}{\displaystyle \sum v_k}\text{converges,}\text{then}{\displaystyle \sum (u_k+v_k)\text{diverges}.}\\ \text{(h)}\text{If}{\displaystyle \sum u_k}\text{and}{\displaystyle \sum v_k}\text{diverges,}\text{then}{\displaystyle \sum (u_k-v_k)\text{converges}.}\\ \text{(i)}\text{If}\text{0}\le {\text{u}}_k\le v_k\text{and}{\displaystyle \sum v_k\text{converges,}\text{then}{\displaystyle \sum u_k\text{converges}\text{.}}}\\ \text{(j)}\text{If}\text{0}\le {\text{u}}_k\le v_k\text{and}{\displaystyle \sum u_k\text{diverges,}\text{then}{\displaystyle \sum v_k\text{diverges}\text{.}}}\\ \text{(k)}\text{If}\text{an}\text{infinite}\text{series converges,}\text{then it converges}\text{absolutely}\text{.}\\ \text{(l)}\text{If an infinite}\text{series diverges}\text{absolutely,}\text{then}\text{it}\text{diverges}\text{. }\end{array}$