Let $f_n(x) = frac{x^2 + nx + 3}{n}$, with $f_n$ defined on $[1,2]$. I must show that $lim_{n to infty} f_n$ converges uniformly to $x$ over $[1,2]$.

I start by defining the set

$$

S = {x : exists lim_{n to infty} f_n(x)}

$$

Then I define the function

$$

f(x) = lim_{n to infty} f_n(x) = x

$$

This seems to prove that ${ f_n(x)}$ converges pointwise. Then I try to determine an $epsilon > 0$, such that

$$

|f_n(x) – f(x)| < epsilon, forall x in S

$$

And here I get stuck. I tried finding a maximum value of $|frac{x^2 + nx + 3}{n} – x|$ using derivatives, but unfortunately this function doesn’t have any around $[1,2]$.

Also, I have no idea how should I find a $N_epsilon$ after finding the $epsilon$ itself. Any ideas or help would be really appreciated. Thanks!

First note that:

$$

left| frac{x^2+nx+3}{n}-xright|=left|frac{x^2+nx+3-nx}{n}right|=frac{x^2+3}{n}le frac{7}{n}

$$

So, given $varepsilon>0$, take $N$ such that $frac{7}{N}< varepsilon$. Then $|f_n(x)-f(x)|<varepsilon$ for every $nge N$ and every $xin [1,2]$.

first we know that $x in [1,2] implies (x^2+3) in [4,7]$

$$|frac{x^2+nx+3}{n} – x| = |frac{x^2+3}{n}| < frac{7}{n} < epsilon$$

hence $n> frac{7}{epsilon}$ suffices hence it converges uniformly