 Solution

The assertion is understood for every $\eta \in \mathbb{R}$ and for a topology in which sequentially closed sets are closed, in particular the norm topology of a Banach space. Suppose first that $f$ is sequentially lower semicontinuous and $x_n\in {\Omega }_{\eta }=\{x:f(x)\le \eta \}$ converges to $x$. Thenso $x\in {\Omega }_{\eta }$ and the sublevel set is closed. Conversely, if all sublevel sets are closed but lower semicontinuity fails, there are $x_n\to x$ and a real $\eta <f(x)$ with a subsequence satisfying $f(x_{n_k})\le \eta$. Closedness of ${\Omega }_{\eta }$ would put $x$ in that set, a contradiction. This proves the closed-sublevel-set characterization of lower semicontinuity.