A question on the liminf of a sequence

Question

Let $(a_n)$ and $(b_n)$ be sequences of nonnegative numbers.
Suppose that $\sum_n a_n=+\infty$ and $\sum_n a_nb_n<+\infty$.
Then it can be shown that $\lim\inf_n b_n=0$.
My question is: What reference can you recommend that contains this and similar results?

Answer 1

The result follows because if $\liminf b_n > 0$ then $b_n > c > 0$ for all sufficiently large $n$ and, hence, $a_n b_n > c a_n$.

The underlying theorems (comparison test, etc.) are found in any standard analysis or advanced calculus book, but it is unlikely that this specific proposition would be displayed except, perhaps, as an exercise.

The insight for making the necessary connections could be gained through practice -- solving many problems in sequences and series. I would begin with one of the books that provide difficult practice problems with answers, such as Problems in Mathematical Analysis by Kaczor and Nowak or Problems and Theorems in Analysis by Polya and Szego.

Also classic references on sequences and series like Theory and Application of Infinite Series by Knopp contain many more theorems and lemmas than modern analysis books.

Addendum

Not inconsistent with what I said above, you will find this in a disguised form in Section II.9 of Theory of Infinite Series by Bromwich (1908 edition):

Let the series $\sum(1/D_n)$ be divergent, then $\sum a_n$ will diverge provided that $\underline{\lim}(a_nD_n) > 0$, both series containing only positive terms.

In short, you would be well served studying this book if you are interested in such results.