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Let $X$ be a vector space, and $\|\cdot\|_1$ and $\|\cdot\|_2$ two different (non-equivalent) Norms on $X.$ Let $(x_n)\subset X$ be a sequence and $x\in X$ such that $\lim_{n\to\infty}\|x_n-x\|_1=0.$ The question is: If $\lim_{n\to\infty}\|x_n-x\|_2>0$ can you say that the sequence $(x_n)$ does not converge (to any other limit) with respect to $\|\cdot\|_2$? Proof?, example? Thanks in advance!

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    In finite dimensions space the norms are equivalent [here](http://www.math.colostate.edu/~yzhou/course/math560_fall2011/norm_equiv.pdf) . However, that is not true in infinite dimensions spaces. See [here](http://www.physicsforums.com/showthread.php?t=451562).2013-04-30

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This was asked and answered in a MathOverflow question, Example of sequences with different limits for two norms. The answer is that there are examples where $(x_n)$ converges in both norms, to different limits.

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    You are right! I'm sorry. Also the example from Mar 23 2011 at 8:44 works in a very simple way. Thanks again!2012-12-06