Proving that $\sum_{n=1}^{\infty} f(a{_{n}})$ converges absolutely

Question

Let $(a_{n})_{n}$ be a sequence of non zero real numbers such that the series $$\sum_{n=1}^{\infty} a{_{n}}$$ converges absolutely. Let $f:\mathbf{R}\rightarrow\mathbf{R}$ be a function with the property that $$\lim_{x\to 0} f(x)/x$$ exists and is finite. Show that $$\sum_{n=1}^{\infty} f(a{_{n}})$$ converges absolutely.

Answer 1

Hint: Denote the limit of $f(x)/x$ as $L$ whenever $x\rightarrow 0$. Try to argue that $\left|f(a_{n})\right|<(|L|+1)|a_{n}|$ for large $n$.

Answer 2

Note that the convergence of $\sum_{n=1}^{\infty} a{_{n}}$ implies

$$\lim_{n\to\infty}a_n= 0$$

Hence $$\lim_{x\to 0} f(x)/x= \lim_{n\to \infty} f(a_n)/a_n= l$$

for $\varepsilon= |l|+1>0$ there exists N such that for $n>N$ we have $$||f(a_n)/a_n|-|l||\le|f(a_n)/a_n-l|<|l|+1\implies |f(a_n)|\le (2|l|+1)|a_n|$$

that is $$\sum_{n=N}^{\infty}|f(a_n)|\le(2|l|+1) \sum_{n=N}^{\infty}|a_n|<\infty\ $$