Let $1\le p<q\le \infty$ and $l^p(\mathbb{R})$ be the set of sequences of real numbers such that the series of absolute values of its terms to the power $p$ converges and $|\cdot|_p$ be the $p$-norm (the same for $l^q(\mathbb{R})$ and $|\cdot|_q$). I would like to know whether it is false or true that if a sequence of elements of $l^p(\mathbb{R})$ converges to an element in $l^p(\mathbb{R})$ using the $q$-norm then the convergence also holds using the $p$-norm (this makes sense because $l^p(\mathbb{R})$ is a subspace of $l^p(\mathbb{R})$).
I did not manage to prove it and the usual counterexamples that look like an harmonic series do not seem to work. I would appreciate some help on how to approach this problem (not a whole solution, please).
