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\flushleft\underline{Homework 10 --- MATH 4590 Spring 2018} \\
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\item Let $f_n \colon [0,2] \rightarrow \mathbb{R}$ be defined by $f_n(x)=\dfrac{x^n}{1+x^n}$. Take the limit as $n \rightarrow \infty$ and explain why it shows that convergence is not uniform.
\item Find a sequence of a functions $f_n \colon [0,1] \rightarrow \mathbb{R}$ which is discontinuous everywhere for every $n$ but converges to a function that is continuous everywhere. (\textit{hint: think about a function we have seen in the past that is discontinuous everywhere})
\item Let $g_n \colon [0,\infty) \rightarrow \mathbb{R}$ be defined by $g_n(x)=\dfrac{e^{-nx}}{n}$. Examine the relationship between the limit function, the derivative of the limit function, $g_n'$, and the limit of $g_n'$. 
\item Let $g_n \colon [0,1] \rightarrow \mathbb{R}$ be defined by $g_n(x)=nx(1-x)^n$. Discuss the convergence of $g_n$ and the convergence of $\displaystyle\int_0^1 g_n(x) \mathrm{d}x$. 
\item Let 
$$f_n(x) = \left\{ \begin{array}{ll}
1, & \quad 0 < x < \dfrac{1}{n} \\
0, & \quad x=0 \text{ or } \dfrac{1}{n} \leq x \leq 1.
\end{array} \right.$$
What is the limit function? Is the convergence uniform?
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