Let $\|.\|_1$ and $\|.\|_2$ be two complete norms on a linear space $X$ such that if a sequence $(x_n)$ converges to $x$ in $(X,\|.\|_1)$ and to $y$ in $(X,\|.\|_2)$, then $x=y$. We have to prove that $\|.\|_1$ and $\|.\|_2$ are equivalent norms.
I know that if there exists $K>0$ such that $\|x\|_1\leq K\|x\|_2$ for all $x\in X$, then by the consequence of open mapping theorem, two norms are equivalent. But how to get this from the available information? Please suggest!