Let $\Omega$ be an open bounded domain and let $p, q>1$. Consider $(x_n)_n\subset W^{1, p}(\Omega)$ and $(y_n)_n\subset W^{1, q}(\Omega)$. Defined $$W= W^{1, p}(\Omega)\times W^{1, q}(\Omega)\quad\mbox{ with } \|(x, y)\|_W = \|x\|_{W^{1, p}} +\|y\|_{W^{1, q}},$$ my question is the following.
If we assume that $$\|(x_n, y_n)\|_W\to +\infty\quad\mbox{ as } n\to +\infty,$$ it can be assumed that at least one between $\|x_n\|_{W^{1, p}}$ and $\|y_n\|_{W^{1, q}}$ goes at $+\infty$? I mean, it is possible to say:
"Assume $\|(x_n, y_n)\|_W\to +\infty$. Thus, without loss of generality, we can suppose that $\|x_n\|_{W^{1, p}}\to +\infty$"?
I hope someone could clarify that.
Thank you in advance!