Let $E$ be some Banach space and $\mathcal{C}_0(\mathbb{R}^n,E)$ the set of continuous functions $\mathbb{R}^n\to E$ vanishing at infinity. I am asked to proof that
$\mathcal{L}_1(\mathbb{R}^n,\lambda_n,E)\cap\mathcal{BC}(\mathbb{R}^n,E){\color{red}\subsetneq}\mathcal{C}_0(\mathbb{R}^n,E)$ and $\mathcal{L}_1(\mathbb{R}^n,\lambda_n,E)\cap\mathcal{BUC}(\mathbb{R}^n,E)\subseteq \mathcal{C}_0(\mathbb{R}^n, E)$.
Am I right in assuming that the red symbol should in fact be "$\not\subseteq$"?
In case $E = \{0\}$, we have $L^1(\mathbb{R}^n,\lambda_n,E) \cap \mathcal{BC}(\mathbb{R}^n,E) = \mathcal{C}_0(\mathbb{R}^n,E) = \{0\}$, so in that case both versions are wrong. If $E\neq \{0\}$, the version with $\not\subseteq$ is correct while the version with $\subsetneq$ remains wrong. To see that $\not\subseteq$ is then correct, consider a continuous real-valued function $g$ with "pyramids" of height $1$ and base length $2^{-k}$ centred at $k\cdot e_1$, zero elsewhere. Then $$g \in L^1(\mathbb{R}^n,\lambda_n,\mathbb{R}) \cap \mathcal{BC}(\mathbb{R}^n,\mathbb{R})\setminus \mathcal{C}_0(\mathbb{R}^n,\mathbb{R})$$ and for every $v\in E\setminus \{0\}$ the function $x \mapsto g(x)\cdot v$ belongs to $$L^1(\mathbb{R},\lambda_n,E) \cap \mathcal{BC}(\mathbb{R}^n,E) \setminus \mathcal{C}_0(\mathbb{R}^n,E).$$
For the case $E = \mathbb{R}$, there are various proofs that an integrable uniformly continuous function belongs to $\mathcal{C}_0(\mathbb{R}^n,\mathbb{R})$, the proofs carry over almost verbatim to the case of general $E$.