Is $\{ f \in C[a,b] | |f(x)| \leq 2+f(x)^2 \}$ a complete metric space with the metric $\rho_{\infty}$ (the sup norm metric)?

Question

Is $\{ f \in C[a,b] | |f(x)| \leq 2+f(x)^2 \}$ a complete metric space with the metric $\rho_{\infty}$ (the sup norm metric)?

So far, I know that continuous functions on a finite interval are complete in the sup norm. So, $|f_n(x) - f_m(x)| \leq \epsilon \implies f_n(x) \rightarrow f(x) \in C[a,b]$, but does $f(x)$ necessarily belong to the above subset?

Answer 1

Hint : Show that $\{f \in C[a,b]:|f(x)| \le 2+|f(x)|^2\}$ is closed in $(C[a,b],||.||_{\infty})$