Quotient of two monotone function Let $(f_n)$ and $(g_n)$ be two positive and monotone increasing sequences of real functions.
Is $\frac{f_n}{f_n+g_n}$ monotone? Does it converge?
My thoughts
$\frac{f_n}{f_n+g_n}=1-\frac{g_n}{f_n+g_n}=1+\frac{(-1)g_n}{f_n+g_n}$. As the numerator $(-1)g_n$ is monotone decreasing and the denominator monotone increasing, the quotient should be monotone decreasing.
If there is $N$ such that $\forall n>N$, the quotient is in some bounded subset (it is eventually bounded), then it converges.
Did I go in the right direction?

Hint: $\frac {f_n} {f_n+g_n}=\frac1  {1+\frac {g_n} {f_n}}$. So this is montone iff  $\frac {g_n} {f_n}$ is monotone. It is easy to construct examples where the ratio of two positive increasing functions is not monotone. (Look at constant functions).  Also, the limit may not exist. I hope you can work out some exmaples.
