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Let $(\lambda_n)_n$ be a sequence of real numbers which converges to $0$ (i.e., is in $c_0$), but is not in $\ell^p$ for any $1\leq p<\infty$, e.g., $\lambda_n=\frac{1}{\log(n+2)}$ for $n=0,1,2\ldots$.

Fix $q$ with $1\leq q<\infty$, and let $(\alpha_n)_n$ be a sequence in $c_0$ but not in $\ell^q$. Can we conclude that the sequence $(\lambda_n\alpha_n)_n$ is also not in $\ell^q$?

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No. For example, $(\alpha_n) = (\frac1{n\log(n+2)})$ is not in $\ell^1$, but $(\lambda_n\alpha_n) = (\frac1{n\log^2(n+2)})$ is in $\ell^1$. Similar examples exist for any $q$.

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