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Consider a strictly increasing sequence $1\leq q_0<q_n<q_{n+1}<q$ such that $q_n\to q$ as $n\to \infty$. Let $B\subset \Bbb R^d$ be a ball, so that $L^{q}(B)\subset L^{q_{n+1}}(B)\subset L^{q_{n}}(B)\subset L^{q_0}(B)$. Consider a sequence $(u_n)_n\subset L^q(B)$ such that $\|u_n\|_{L^{q_n}(B)}=1$ and $u_n\to u$ in $L^{q_0}(B)$.

I want to prove or disprove that $\|u\|_{L^{q}(B)}=1$. Note that by Fatou's lemma, we get $$\|u\|_{L^{q}(B)}\leq \liminf_{n\to\infty}\|u_n\|_{L^{q_n}(B)}=1.$$

Consider a strictly increasing sequence $1\leq q_0<q_n<q_{n+1}<q$ such that $q_n\to q$ as $n\to \infty$. Let $B\subset \Bbb R^d$ be a ball, so that $L^{q}(B)\subset L^{q_{n+1}}(B)\subset L^{q_{n}}(B)\subset L^{q_0}(B)$. Consider a sequence $(u_n)_n\subset L^q(B)$ such that $\|u_n\|_{L^{q_n}(B)}=1$ and $u_n\to u$ in $L^{q_0}(B)$.

I want to prove or disprove that $\|u\|_{L^{q}(B)}=1$. Note that by Fatou's lemma, we get $$\|u\|_{L^{q}(B)}\leq \liminf_{n\to\infty}\|u_n\|_{L^{q_n}(B)}=1.$$

Addendum: I forgot the assumption that $d>q.$

The answer by @FedorPetrov gives a nice counter example when q>d.

Consider a strictly increasing sequence $1\leq q_0<q_n<q_{n+1}<q$ such that $q_n\to q$ as $n\to \infty$. Let $B\subset \Bbb R^d$ be a ball, so that $L^{q}(B)\subset L^{q_{n+1}}(B)\subset L^{q_{n}}(B)\subset L^{q_0}(B)$. Consider a sequence $(u_n)_n\subset L^q(B)$ such that $\|u_n\|_{L^{q_n}(B)}=1$ and $u_n\to u$ in $L^{q_0}(B)$.

I want to prove or disprove that $\|u\|_{L^{q}(B)}=1$.

Consider a strictly increasing sequence $1\leq q_0<q_n<q_{n+1}<q$ such that $q_n\to q$ as $n\to \infty$. Let $B\subset \Bbb R^d$ be a ball, so that $L^{q}(B)\subset L^{q_{n+1}}(B)\subset L^{q_{n}}(B)\subset L^{q_0}(B)$. Consider a sequence $(u_n)_n\subset L^q(B)$ such that $\|u_n\|_{L^{q_n}(B)}=1$ and $u_n\to u$ in $L^{q_0}(B)$.

I want to prove or disprove that $\|u\|_{L^{q}(B)}=1$. Note that by Fatou's lemma, we get $$\|u\|_{L^{q}(B)}\leq \liminf_{n\to\infty}\|u_n\|_{L^{q_n}(B)}=1.$$

Consider a strictly increasing sequence $q_0<q_n<q_{n+1}<q$ such that $q_n\to q$ as $n\to \infty$. Let $B\subset \Bbb R^d$ be a ball, so that $L^{q}(B)\subset L^{q_{n+1}}(B)\subset L^{q_{n}}(B)\subset L^{q_0}(B)$. Consider a sequence $(u_n)_n\subset L^q(B)$ such that $\|u_n\|_{L^{q_n}(B)}=1$ and $u_n\to u$ in $L^{q_0}(B)$.

I want to prove or disprove that $\|u\|_{L^{q}(B)}=1$.

Consider a strictly increasing sequence $1\leq q_0<q_n<q_{n+1}<q$ such that $q_n\to q$ as $n\to \infty$. Let $B\subset \Bbb R^d$ be a ball, so that $L^{q}(B)\subset L^{q_{n+1}}(B)\subset L^{q_{n}}(B)\subset L^{q_0}(B)$. Consider a sequence $(u_n)_n\subset L^q(B)$ such that $\|u_n\|_{L^{q_n}(B)}=1$ and $u_n\to u$ in $L^{q_0}(B)$.

I want to prove or disprove that $\|u\|_{L^{q}(B)}=1$.