Im reading a paper, where I find this problem:
Let $f, g$ two function such that from a Banach space $\mathbb R$ into itself:
- $f$ is continuous, and $g$ is lower semi-continuous;
- $(x_n)_{n\in \mathbb N}$ which converges to x.
- $f(x_n)\leq f(x_n)-g(x_n),\:\forall n\in \mathbb N$.
In the passage, the author writes:
Taking the limsup in the above inequality, we have:
$$\begin{matrix} \limsup _{n \rightarrow \infty} f(x_n) &\leq &\limsup _{n \rightarrow \infty} f(x_n)-\liminf _{n \rightarrow \infty} g(x_n)\\ \text{ Then }\\ f(x) &\leq &f(x)-g(x) \end{matrix}$$
I think that ther is something wrong about this, any help!