# Discuss the convergence and uniform convergence of the sequence of functions, $f_n: [0, \infty) \to \mathbb{R}:$

$$f_n(x)=\frac{nx^2}{1+nx^2}$$

In the case of non-uniform convergence, find subintervals on which the convergence is uniform and prove your conjecture.

I'm thinking that because both the numerator and denominator are increasing at the same rate as n approaches infinity, then $$lim_{n \to \infty}f_n(x)=1$$ for all $$x \in (0, \infty]$$.

For the uniform convergence piece, I'm thinking that $$f_n(x)$$ converges uniformly on the interval $$[\eta,\infty)$$, but i'm kind of confused on that part. The definition of uniform convergence, geometrically, says that if we make a vertical "collar" of radius $$\epsilon$$ around the graph of the limit function $$f$$, the graphs of all functions in the sequence after the $$N$$th one must lie within the "collar". But i feel like no matter how large you make $$\epsilon$$, you can fit in any $$f_n(x)$$.

What I'm thinking for the uniform convergence proof:

Fix $$\eta$$ such that $$0<\eta<\infty$$. Claim that $$(f_n)$$ converges uniformly on $$[\eta, \infty)$$ to $$f(x)=1$$. Fix $$\epsilon>0$$ and choose $$N \in \mathbb{N}$$ such that, $$N> ?$$. Not sure what to pick for $$N$$.

I'm trying to choose $$N$$ based on, $$d(f_n(x),f(x))<\epsilon$$, but i'm struggling to find something to use.

• Check Dini's theorem (en.wikipedia.org/wiki/Dini%27s_theorem). Using it, I think you can argue that on any compact interval that does not include zero the convergence is uniform (and implied by pointwise). – Jan Dec 20 '18 at 2:03

$$\displaystyle f_n(x)=\frac{nx^2}{1+nx^2}=\dfrac{x^2}{\frac{1}{n}+x^2}\to\begin{cases}0 &x=0\\1 &x\neq0\end{cases}$$
• I'm confused by what you are telling me here... how does $\frac{nx^2}{1+nx^2} = \frac{x^2}{\frac{1}{n}+x^2}$? – big_math_boy Dec 20 '18 at 2:02
• dividing the numerator and denominator by $n$, $n\neq0$. – Yadati Kiran Dec 20 '18 at 2:03
• No, $f_n\to f$ uniformly for $x\neq0$ i.e. $x\in(0,a),a\in\mathbb{R}$. – Yadati Kiran Dec 20 '18 at 2:08
• What $N$ do you recommend I use to prove this? – big_math_boy Dec 20 '18 at 15:53