I just came across a situation where $\mathbb{H}$ is a real Hilbert space and $\lbrace x_n \rbrace $ is a sequence in $\mathbb{H}$ that converges to some $\overline{x}$. Also, the sequence satisfies the inequality $\langle f(x_n),y-x_n \rangle \geq{0}$, for all $y\in{C}$, where $C$ is a nonempty, closed and convex subset of $\mathbb{H}$. Clearly, we can take the limit in the above inequality such that $\langle f(\overline{x}),y-\overline{x} \rangle \geq{0}$ is true, for all $y\in{C}$. My question is why is this still true for all $y\in{C}$, after taking the limits?
edit: Thanks to Euclid for the reminder that $f$ is a continuous function.