# Tag Info

### Is $\mathbb{R}\cup\{-\infty,+\infty\}$ the categorical co-completion of $\mathbb{Q}$

The free cocompletion of any category is its category of presheaves. The category of presheaves on $\mathbb Q$ is the category of families of sets, contravariantly indexed by rational numbers. This is ...
1 vote

### Is "uniform continuity theorem" provable without using sequences?

You can't do it like that. Indeed, if such an argument worked, you could use the same reasoning to prove that any function $\mathbb{R} \rightarrow \mathbb{R}$ is uniformly continuous on $\mathbb{R}$ (...
1 vote

If $f$ is differentiable in $a$ then you know that for $v = (v_1,v_2)$ you should have $$\frac{\partial f}{\partial v} (a)= (\nabla f (a)) \cdot v = v_1 \frac{\partial f}{\partial e_1}(a) + v_2 \... 1 vote Accepted ### Prove \exp(\int^x_0 \frac{U(t)}{t} dt) is reguarly varying (Extreme Values, Regular Variation and Point Processes) Let's denote f(x) = \exp \left(\int\limits_0^x \frac{U(t)}{t} {\rm d}t\right). Following the given theorem, one needs to prove that \int\limits_0^1 \ln \left(\frac{f(x)}{f(\lambda x)}\right) {\rm d}... 1 vote ### The set X := [1, 2] \cup [3, 4] is not connected Openness and closedness depends on the ambient space. In the very common ambient space of the real numbers, [1,2] is a non-open closed subset, as there does not exist any r > 0 such that we ... 1 vote ### The set X := [1, 2] \cup [3, 4] is not connected I think you can agree that in any metric space (X,d), the open balls B_r(x) with radius r centered at x are open. In fact, by definition a subset Y\subseteq X of a metric space is open if ... 1 vote ### Can this proof of the second MVT for integrals be salvaged? It seems unlikely that we can extend this argument to cases where f changes sign on [a,b]. We can easily construct examples where \int_a^b fg does not lie between A and B. Let g(x)=x, f(x)... 1 vote Accepted ### A question about the d'Alembert ratio test for convergence. Suppose we do have some sequence \{ a_n \} that fulfills the property as described, i.e. \lim_{n \rightarrow \infty}|\frac{a_{n + 1}}{a_n}| = L > 1. With this assumption, we can choose some \... 1 vote ### Local minima and derivative transfer everything except for o(h) on the LHS, use the definition of o(h) - particularly the definition of a limit (epsilon delta definition), put epsilon := (1/2)|f′(x∗)|. Normal people use epsilon ... 1 vote Accepted ### Can the convergence of inner products be 'closed'? What follows is an example that the uniform convergence on K does not suffice. Let \{e_j\}_{j=1}^\infty be an orthonormal basis. Consider$$v_n=\sum_{j=1}^n 2^{-j}e_j,\quad v=\sum_{j=1}^\infty 2^{-...

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