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Два определения линейной сходимости получаются неэквивалентны, так как, например, последовательность из 12 задачи ДЗ удовлетворяет первому определению, но не удовлетворяет второму:
$$
r_k =\begin{cases} \frac{1}{2^k}, & \mbox{если } k\mbox{ нечетно} \ \frac{1}{3^{2k}}, & \mbox{если } k\mbox{ четно} \end{cases}
$$
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Два определения линейной сходимости получаются неэквивалентны, так как, например, последовательность из 12 задачи ДЗ удовлетворяет первому определению, но не удовлетворяет второму: