Thomas E.
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 Jun 2 answered Two problems related to continuity of a metric from Munkres' topology book Jun 2 awarded Excavator Jun 2 reviewed Approve If I invert the argument, should I invert the constants in the equation? Jun 2 reviewed Approve Berlekamp's algorithm Jun 2 reviewed Reject Ordinal arithmetic and limit ordinals Jun 2 revised Monotone Convergence Theorem for non-negative decreasing sequence of measurable functions another tag. Jun 2 revised Monotone Convergence Theorem for non-negative decreasing sequence of measurable functions Hi. I fixed one point at your answer, namely the claim about the limit (crucial typo there: you wrote f_n instead of f). And also, +1 ! Jun 2 reviewed Approve Limit of a function involving trigonometric exponents. Jun 2 comment What is the topology here with three elements in sets? @Topology: The complement of $\{b\}$ is $\{a,c\}$, which is neither an element of $\tau$, nor $\tau$ without $\{b\}$ in it. Jun 2 answered What is the topology here with three elements in sets? Jun 2 revised How to show that $\lambda(E)=1$?? edited tags Jun 2 answered How to show that $\lambda(E)=1$?? Jun 2 comment Compactness of a group with a bounded left-invariant metric Why is this left invariant? Maybe it's obvious but I'm not seeing it. Jun 1 reviewed Approve continuity of product path--pasting lemma Jun 1 reviewed Approve Does the square of uniform distribution have density function? Jun 1 comment Prove the dominated convergence for $f_n(x)=\frac{x^n}{x^2+3x+2}$ @Lucas: The conclusion is that $\lim_{n\to\infty}\int_{0}^{1}f_{n}(x)=\int_{0}^{1}f(x)$. And $f(x)=0$ if $x\in [0,1)$ and $f(1)=\frac{1}{6}$. This limit function is defined on the whole interval $[0,1]$ and is discontinuous. The reason why $\int_{0}^{1}f(x)=0$ requires measure theory: the set $\{1\}$ has measure zero and thus its contribution to the integral $\int_{0}^{1}f(x)$ is negligible. To fully understand this, a good place to start is to google "Lebesgue measure" and "Lebesgue integral". Jun 1 comment Prove the dominated convergence for $f_n(x)=\frac{x^n}{x^2+3x+2}$ @lucas: Yes that's true. You can ignore the comment about measure zero: I thought you we're dealing with integration through measure theory, but you can also view the integral as a Riemann integral. That's fine too. Jun 1 comment Prove the dominated convergence for $f_n(x)=\frac{x^n}{x^2+3x+2}$ @lucas: The point wise convergence to zero happens for all $x\in [0,1)$. At $x=1$ the limit is not zero, but the single point $x=1$ has measure zero and thus does not contribute to the value of the integral. Jun 1 comment Prove the dominated convergence for $f_n(x)=\frac{x^n}{x^2+3x+2}$ @Lucas: I edited the answer and included that part too. For further reference for dominated convergence theorem, the wikipedia article has a lot of information: en.wikipedia.org/wiki/Dominated_convergence_theorem Jun 1 revised Prove the dominated convergence for $f_n(x)=\frac{x^n}{x^2+3x+2}$ added 144 characters in body