Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 2 / 12E / a / iii / Solution

SupposeIf $(v_n)$ is a Cauchy sequence in $\Vert \cdot \Vert$, thenConversely, $\Vert v_n-v_m\Vert \le c^{-1}\Vert v_n-v_m{\Vert }^{\prime }\to 0$. Hence the sequence is Cauchy for one norm exactly when it is Cauchy for the other.