Converse of Dirichlet test: a result about existence I am wondering whether the following converse to the Dirichlet's test of convergence is true.

Let $\lambda_n$ be a sequence of complex numbers such that $\sum_{k=1}^n \lambda_k$ is unbounded as $n \to \infty$. Then there exists a sequence $x_n \to 0$ such that $\sum \lambda_n x_n$ diverges.

There is a delicate balance between $\lambda$ and $x$, which makes it really hard to prove the above result or find a counterexample.
Is the statement above true?
In fact, if a counterexample exists, then there might be some possibilities of strengthening the Dirichlet test to a stronger result.
 A: First step : Let's begin with the case where $\lambda_k \in \mathbb{R}_+$ for all $k$. In that case, the sequence $(S_n)_{n \geq 1}$ defined for all $n \geq 1$ by $$S_n = \sum_{k=1}^n \lambda_k$$ is increasing and diverges to $+\infty$. Considering $$x_n = \frac{1}{S_n}$$
You have for all $N, p \in \mathbb{N}^*$,
$$\sum_{k=N}^{N+p} \lambda_k x_k = \sum_{k=N}^{N+p} \frac{\lambda_k}{S_k} \geq \sum_{k=N}^{N+p} \frac{\lambda_k}{S_{N+p}} = \frac{S_{N+p}-S_N}{S_{N+p}} = 1 - \frac{S_N}{S_{N+p}}$$
which tends to $1$ as $p$ tends to $+\infty$, so you see that the sequence $\sum \lambda_k x_k$ is not Cauchy, so it does not converge.
Second step : Now we come back to the general case, where $\lambda_k \in \mathbb{C}$. We can suppose without loss of generality that for every $k \geq 1$, one has $\lambda_k \neq 0$ (just removing the zero terms from the sums).
First, notice that
$$\sum_{k=1}^n |\lambda_k| \geq \left|\sum_{k=1}^n \lambda_k \right|$$
so because $\sum \lambda_k$ is supposed to be unbounded, then $\sum |\lambda_k|$ diverges to $+\infty$. By the first case applied to $|\lambda_k|$, there exists a sequence $(x_n)$ which tends to $0$, such that $\sum |\lambda_k|x_k$ diverges. Define $(y_n)$ by
$$y_n = \frac{\overline{\lambda_n}}{|\lambda_n|}x_n$$
Of course $|y_n|=|x_n|$ so $(y_n)$ tends to $0$, and $$\sum \lambda_k y_k = \sum \lambda_k \frac{\overline{\lambda_k}}{|\lambda_k|}x_k = \sum |\lambda_k|x_k$$
which diverges by definition of the sequence $(x_n)$.
