# Tagged Questions

Theoretical foundations of calculus: limits, convergence of sequences, construction of the real numbers, least upper bound property, and related analysis topics such as continuity, differentiation, and integration through the fundamental theorem of calculus.

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### Expressing $\Bbb N$ as an infinite union of disjoint infinite subsets.

The title says it. I thought of the following: we want $$\Bbb N = \dot {\bigcup_{n \geq 1} }A_n$$ We pick multiples of primes. I'll add $1$ in the first subset. For each set, we take multiples of some ...
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### Can you find a formula for the supremum of an infinite collection of nonempty sets each of which is bounded above?

I believe the supremum of a collection of finite sets which are all nonempty and bounded above is simply the maximum of the set containing the supremums of each set, but can you find the supremum of a ...
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### If $(x_n)_{n\in\mathbb{N}}$ and $(y_n)_{n\in\mathbb{N}}$ are convergent in $\mathbb{R}^m$ , then $\|x_n - y_n\|$ is convergent in $\mathbb{R}$

To demonstrate this proposition I use the following: Definition $\quad$ $(x_n)_{n\in\mathbb{N}}\subset \mathbb{R}^m$ converges to $x\in\mathbb{R}^m$ if and only if $\forall \varepsilon >0$ there ...
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### Showing $\exists$ unique $x>0$ such that $x^{2} = k$

I'm trying to work through this proof, and have hit a stumbling block. This is what I have so far: Let $0<k \in \mathbb{R}$, where $E = \{ x \in \mathbb{R} | x^{2} < k\}$. First, we must show ...
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### Proving or disproving continuity of $f(x+a)=\frac{1+f(x)}{1-f(x)}$

Let $f$ be defined $\forall x\in\mathbb{R}$ and let $$f(x+a)=\frac{1+f(x)}{1-f(x)}$$ hold for some $a\in\mathbb{R}^{+}$ and $\forall x\in\mathbb{R}$. Show that $f$ is periodic with a period of $4a$ ...
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### “alternate definition” of continuity

I was trying to find a definition of the continuity of a function that fits better the unrirgorous definition that "it can be drawed with a pencil in a single stroke" and I came up with this: a ...
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### Normalization in $L^{p}$ and $L^{q}$

Given a function $f$ in $L^{p}\cap L^{q}$ where $0<p,q<\infty$, can $f$ always be normalized such that $\| f \|_p=\| f \|_q=1$?
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### Confused by Monotone class theorem for functions

Monotone Class Theorem has two types. One is Monotone class theorem for sets and the other for functions. I have no doubt for sets. Here is a reference of definition of Monotone Class Theorem for ...
Let $\mu$ be a $\sigma$-finite measure on $(\mathbb{R},\text{Borel sets on }\mathbb{R})$, which is absolutely continuous with respect to the Lebesgue measure. Then, is it true that, $\mu(K) < \... 1answer 90 views ### Proof that a number and its multiplicative inverse have the same sign I am trying to prove that if a number$a > 0$, then its multiplicative inverse$a^{-1}$is also$>0$. What I have done thus far is this: Using the trichotomy property for the real numbers, I ... 1answer 373 views ### let$A_1, A_2, A_3, \dots$be a collection of nonempty sets, each of which is bounded above.$(a)$Find a formula for$\sup(A_1\cup A_2)$. Let$A_1, A_2, A_3,\dots$be a collection of nonempty sets, each of which is bounded above.$(a)$Find a formula for$\sup(A_1 \cup A_2)$. Extend this to supremum of a collection of$n$sets$A_1,...
I have the following problem: If $v$ is a value of a continuous function $f:[a,b] \rightarrow \mathbb{R}$, use the least upper bound property to prove that there are smallest and largest $x \in [a,b]$ ...