Hey, I've got a real analysis question here. Any help would be greatly appreciated:
Let f(x) = {1 if x = 1/n for some n Є N
{0 otherwise
Show that f is integrable on [0,1] and compute
$\displaystyle
\int_1^0 {f}
$
Hey, I've got a real analysis question here. Any help would be greatly appreciated:
Let f(x) = {1 if x = 1/n for some n Є N
{0 otherwise
Show that f is integrable on [0,1] and compute
$\displaystyle
\int_1^0 {f}
$
consider this partition let $\displaystyle \epsilon >0 $
P\fix N such that $\displaystyle \frac{1}{N} < \frac{\epsilon}{4}$
$\displaystyle p_n=\left( \frac{1}{n}-\frac{\epsilon}{2^{n+1}},\frac{1}{n}+\frac{\epsilo n}{2^{n+1}} \right)$
for n=2 to n=N
notice that $\displaystyle \sum_{n=2}^{N}\left(\frac{1}{n}+\frac{\epsilon}{2^ {n+1}}-\left( \frac{1}{n}-\frac{\epsilon}{2^{n+1}}\right) \right)=\sum_{n=2}^{N}\frac{\epsilon}{2^{n}}<\frac {\epsilon}{2}$
See if you can put it together from here
Good luck.