Second rule of comparison in math series

[Alem2000]

I can't understand this $$\sum{A_n}\leq \sum{B_n}$$ having said this

if $$\sum{B_n}$$ converges so does $$\sum{A_n}$$, okay that

makes perfect sense but then the second rule of comparison is if $$\sum {A_n}$$ diverges then so does $$\sum{B_n}$$ diverges too...can

anyone tell me how that makes sense? A proof maybe..?

 

[matt grime]

General result, if P implies Q, then not Q implies not P.

replace P with sum Bn converges and Q sum An converges.

 

[HallsofIvy]

Or:

Given $$\sum{A_n}$$ does NOT converge.

Now assume $$\sum{B_n}$$ DOES converge. Using the theorem you said "makes perfect sense", what does that tell you about $$\sum{A_n}$$
?