I apologise if the title is confusing or misleading.
- Let $(f_i)_{i\in\mathbb{N}}$ be a sequence of continuous functions converging pointwise to a continuous function $f:\mathbb{R}^n\to\mathbb{R}$ from below i.e. $f_i(x) \leq f(x)$ and $\lim_{i\to\infty}f_i(x) = f(x)$ for all $x\in\mathbb{R}^n$ and $i\in\mathbb{N}$.
- Suppose $f_i$ is bounded below by an affine linear function $l_i(x) = b_i + c^T_i x$ for some constants $b_i\in\mathbb{R}$ and $c_i\in\mathbb{R}^n$. That is $l_i(x) \leq f_i(x)$ for all $i\in\mathbb{R}^n$ and $i\in\mathbb{N}$.
Question: Is it true that $\sup_{i\in\mathbb{N}} |b_i| < \infty$ and $\sup_{i\in\mathbb{N}} \|c_i\| < \infty$ ??? Then does this imply that $l_i$ converges pointwise?
My thoughts: I think that since $\sup_{i\in\mathbb{N}} f_i(x) < f(x) < \infty$ for all $x\in\mathbb{R}^n$, then $\sup_{i\in\mathbb{N}} l_i(x) < f(x) < \infty$ for all $x\in\mathbb{R}^n$. Therefore, if either $\sup_{i\in\mathbb{N}} |b_i|$ or $\sup_{i\in\mathbb{N}} \|c_i\|$ is unbounded, then there exists an $x_0$ such that $\sup_{i\in\mathbb{N}} l_i(x_0)$ is unbounded. Therefore, we have a contradiction. Is this right?
Counterexample for the first question: Let $f(x)$ and all $f_k(x)$ be the function $|x|^2.$ Set $l_k(x) = kx_1-k^2.$