Suppose two norms $|| \cdot||_1$ and $|| \cdot||_2$ for a linear space $X$ preseves boundeddness of sets. i.e $S \subseteq X$ is bdd wrt $|| \cdot||_1$ if and only if $S$ is bdd wrt $|| \cdot||_2$.
Now can we prove that
if $\{x_n\}$ is a sequence in $X$ and $x \in X$, $\{x_n\}$ converges to $x$ in $|| \cdot||_1$ if and only if $\{x_n\}$ converges to $x$ in $|| \cdot||_2$
My Attempt:
If we take a sequence $\{x_n\}$ converges to $x$ in $|| \cdot||_1$, then for any $\epsilon \gt0$ there is a $N\in \mathbb Z^+$ s.t $||x_n-x||_1\lt\epsilon$ for $n\gt N$.
thus the set $S=\{x_n-x:n\gt N \}$ is bdd in $||\cdot||_1$ by $\epsilon$. then by hypothesis $S$ is bdd in $||\cdot||_2$. but from here I cannot proceed. Any hint is appreciated.
Edit: My original question askes me to prove the following conditions are equivalent
1.$|| \cdot||_1$ and $|| \cdot||_2$ are equivalent
2.Boundedness of sets is preserved
3.Convergent of sequences is preserved
4.Sequences being Cauchy is preserved.
it is easy to prove 1 $\implies $ 2 and 3 $\implies 4 \implies $ 1.