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In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (i.e., eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For a set, they are the infimum and supremum of the set's limit points, respectively. In general, when there are multiple objects around which a sequence, function, or set accumulates, the inferior and superior limits extract the smallest and largest of them; the type of object and the measure of size is context-dependent, but the notion of extreme limits is invariant...
Let's say I have a (uniformly) continuous functions $f:[a,b] \to \mathbb{R}$ and an arbitrary function $h:\mathbb{R}^2 \to [a,b]$ such that
$$
\lim_{t\to 0} ~h(t,x) = h(0,x) = x
$$
for all $x$.
I would like to be able to conclude that
$$
\lim_{t\to 0}\int_a^bf(h(t,x))dx = \int_a^bf(x)dx.
$$
...
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Fix any $a$ in $(0, \pi)$. For each $k$ in $\mathbb{N}$ define the sequence $s_{k}=\int_{a}^{\pi}(\sin k x) / k x d x$. Prove that
$\lim _{k \rightarrow \infty} s_{k}=0$
In order to tackle such exercises, we have to find some $\delta$ for each $\epsilon$ in which whenever $k \geq \delta$ holds, t...
Let $ N \unlhd G $ and suppose $G\diagup N $ is abelian. Let C be the group of linear
characters of $G\diagup N $ so that C acts on $Irr(G)$ by multiplication.
Let $θ \in Irr(N)$. Show that
$θ^G=f \sum x_i $ where $f$ is integer and $x_i \in Irr(G)$
Try: I tried using gallaghers theorem but co...
