I have a homework problem which I feel should be simple but is actually surprisingly tricky. This is why I love math sometimes....
Let $X$ be a normed linear space. Suppose $\|\cdot\|_1$ and $\|\cdot\|_2$ are two norms on $X$ such that $x_n\to x_0$ in $\|\cdot\|_1$ if and only if $x_n\to x_0$ in $\|\cdot\|_2$.
Show that $\|\cdot\|_1$ and $\|\cdot\|_2$ are equivalent.
What I have done so far is show that if the topologies induced by $\|\cdot\|_1$ and $\|\cdot\|_2$ are equal, then the norms are equivalent. So I feel that this is only a partial solution because I have failed to show that the given assumption about convergence implies that the induced topologies by each norm are equal. I think this is a well known fact but I should still prove it.
I start by taking a $\|\cdot\|_1$-neighborhood $U$ of $0$ in $X$. Now I would need to show that for any $x\in U$, there is a $\|\cdot\|_2$-neighborhood $V$ of $x$ such that $V\subset U$. This would establish that every $\|\cdot\|_1$-open set is $\|\cdot\|_2$-open. The other direction would be exactly the same. Now I feel pretty dense here (not in the topological sense either) but I can't see how to apply the sequence condition. Any hints or advice on how to do this?