I have no idea on part (b) how to prove or disprove on such set unbounded set R ?
Let {a_{n}},{b_{n}}, {c_{n}} be three sequence converges to a, b, c respectively.
(a) Let B > 0 be a real number. Let f_{n}(x) = a_{n} + b_{n}x + c_{n}x^{2}. Show that {f_{n}} converges uniformly to f(x) = a + bx + cx^{2} on [-B,B].
(b) Prove or disprove that {f_{n}} converges uniformly to f(x) on R.
Let $a_n = c_n = 0$ for all $n$. Let $b_n = 1+\dfrac{1}{n}$. Hence, $f_n(x) = \left(1+\dfrac{1}{n}\right)x$ and $f(x) = x$. Hence, $|f_n(x) - f(x)| = \dfrac{|x|}{n}$. Given any $\varepsilon>0$, if $|x|>n\varepsilon$, then $|f_n(x)-f(x)|>\varepsilon$. This is a proof by counterexample (it is not uniformly convergent).