If $y_{n+1} - \lambda \bar y_n \rightarrow 0$ and $\lambda$ has real part $< 1$ then $y_n \rightarrow 0$.

I was going through the article https://doi.org/10.2307/3215382 from Raul Gouet and stumbled upon one of his appendix lemmas:

I think I understood everything except for the convergence towards zero in the last line. I was trying to use kroneckers lemma, but this didn't work because I can't know if $\sum_{k = 1}^n \frac{\delta_i}{i}$ converges. Could someone please explain to me this last step or alternatively provide another proof (in this case, we can also use the facts that $|\lambda| \leq 1$ and that $(y_n)$ is bounded as the usecase of this lemma provides these facts as well)? Also, I was wondering if the statement is even true at all. However, if it's not true then I think that the whole article doesn't work out.