Proving $\sum_{n \geq 1} \frac{a_n}{1+ a_n}$ diverges provided $\sum a_n$ diverges and $(a_n)$ is decreasing and nonegative

Question

Could someone tell me why this might a wrong solution? The solution looked nothing like mine

Let $t_n$ be the partial sums of $\sum_{n \geq 1} \frac{a_n}{1+ a_n}$

$$t_n = (1-\frac{1}{1+a_1}) + (1-\frac{1}{1+a_2}) + \dots + (1-\frac{1}{1+a_n}) \\ \geq (1-\frac{1}{1+a_n}) + (1-\frac{1}{1+a_n}) + \dots + (1-\frac{1}{1+a_n}) \\ \geq n - \frac{n}{1+a_n} \geq n$$

So $n \to \infty$ shows that the partial sums are unbounded

Answer 1

If $a_n$ is decreasing and positive, then there exists $M>0$ such that $0<a_n<M$ for all $n$.

Then $\frac{a_n}{1+a_n}\ge \frac{a_n}{1+M}$ and therefore the series $\sum \frac{a_n}{1+a_n}$ diverges.

Answer 2

The final inequality that you've typed above is incorrect: $n-\frac{n}{1+a_n}\leq n$ since $\frac{n}{1+a_n}\geq 0$ as noted in the comments.