# Investigate whether the sequence of functions $(f_n)$ defined by $f_n=\frac{nx^3}{7+nx^2}$ is uniformly convergent as follows.

For any $$n \in \Bbb N$$, let $$f_n:\Bbb R \to \Bbb R$$ be a function defined by $$f_n(x)=\frac{nx^3}{7+nx^2}, \quad \forall x \in \Bbb R.$$ Investigate whether the sequence $$(f_n)$$ is uniformly convergent on $$\Bbb R$$.

attempt: I've shown that $$f_n \to f$$ on $$\Bbb R$$, where $$f:\Bbb R \to \Bbb R$$ is a function defined by $$\begin{equation*} f(x)= \begin{cases} 0, \quad x=0 \\ x, \quad x\ne 0. \end{cases} \end{equation*}$$

Notice that for any $$n \in \Bbb N$$ and for any $$x \in \Bbb R \setminus \{0\}$$, we have $$\frac{x^2}{(7+nx^2)^2} < \frac{x^2}{7+nx^2} < \frac{x^2}{nx^2} \implies \frac{|x|}{7+nx^2}<\frac{1}{\sqrt{n}}.$$ I claimed that $$(f_n)$$ is converges uniformly on $$\Bbb R$$ to $$f$$. To this end, let $$\epsilon>0$$ be arbitrary. Choose $$K \in \Bbb N$$ with $$K>\frac{49}{\epsilon^2}$$ such that for any $$n \in \Bbb N$$ with $$n \ge K$$, we have $$|f_n(x)-f(x)| = \left|\frac{nx^3}{7+nx^2}-x \right| = \frac{7|x|}{7+nx^2} < \frac{7}{\sqrt{n}} \le \frac{7}{\sqrt{K}}< \epsilon,$$ for any $$x \in \Bbb R\setminus \{0\}$$. Hence, $$(f_n)$$ is converges uniformly on $$\Bbb R$$ to $$f. \qquad \Box$$

Does it correct? On the other hand, I think if the sequence of functions converges to a piecewise function, then the sequence is not uniformly convergent. I confusing on it. What's is the correct approach? Thanks in advanced.

• note that $f$ is continuous
– ling
May 14 at 1:26
• @ling So? Does it correct or not? May 14 at 1:28
• I think it is correct. No need for f to be piece-wise defined. f(x) = x. May 14 at 1:31
• Your approach is correct . May 14 at 1:59