Do sequence of converging affine linear functions always converge to an affine linear function?

Question

1) Suppose $(f_i(x))_{i\in\mathbb{N}}$ is a sequence of real valued affine linear functions in x. Suppose $f_i(x) \to f(x)$ for all $x\in R^n$. Is $f$ an affine linear function?

2) Suppose $f_i\to f$ pointwise as above where $f$ is an affine linear function. If $a_i$ and $b_i$ is the intercept and slope of $f_i$, is it true that both $\sup |a_i| < \infty$ and $\sup \|b_i\| < \infty$?

Answer 1

Let $(e_1,...,e_n)$ be a basis of $\mathbb{R}^n$. Write $f_p(x)=a_1^px_1+...+a_n^px_n+a_0^p$, you have $lim_p(f_p(0)=lim_pa_0^p=a_0$, $(f_p(e_i))$ converges implies that $(a_i^p)$ converges towards $a_i$. Let $f(x)=a_1x_1+...+a_nx_n+a_0$, $|f_p(x)-f(x)|\leq |a_1^p-a_1||x_1|+...+|a_n^p-a_n||x_n|$ implies $lim_pf_p(x)=f(x)$.