# Tagged Questions

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### Proof of The First Fundamental Theorem of Calculus: Integrating Derivatives

We're given the proof of the First Fundamental Theorem of Calculus and asked to show that it is necessary to assume that the function $f(x):[a,b] \to \Bbb{R}$ is continuous at the endpoints of the ...
4answers
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### Prove that $a \equiv b \pmod{m_1m_2}\implies a \equiv b \pmod {m_1}$

So, I have this problem: if $$a \equiv b \mod(m_1m_2)$$ then (show) $$a \equiv b \mod(m_1)$$. I have to do a proof, but I have no idea where to begin the proof. Can someone help? Proof (Edit): ...
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### Proof using properties of an isosceles or right-angle triangle

Given a triangle $ABC$ with sides $AB=BC$ and angle$\angle B=100^\circ$, prove that $$a^3 + b^3 = 3a^2b$$ where $a=AB=BC$ and $b=AC$, I have tried to use simultaneously the sine and cosine rules as ...
2answers
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### Proof of natural log identities

I need to prove a few of the following identities from a real analysis perspective- this means I do not have access the $\ln e^2 = 2$ type definition of the log function. I am developing the log ...
1answer
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### Proving the limit of a function of a sequence is equal to the function of the limit of that sequence

Suppose $f$ is a continuous function at $x = c$ in $[a,b]$. Prove that for any sequence ${x_n}$ in $[a,b]$ converting to $c$, the sequence $\{f(x_n)\}$ converges to $f(c)$. That is,  ...
2answers
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### Proving the continuity of a difference of functions.

Prove that if $f$ and $g$ are continuous at $x=a$, then $(f-g)$ is continuous at $x=a.$ I have $|f(x)-f(a)-g(x)+g(a)| = |(f(x)-g(x))-(f(a)-g(a))|$ so far. I wanted to use the triangle inequality on ...
1answer
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### Proof that there is only one homomorphism from Z to Z/nZ

Could anyone help me (even just a start) to prove this ? Homomorphism is a new notion for me and I have to confess that I am a bit lost, I don't know how to start. Thanks in advance
1answer
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### Proving a Property of a Set of Positive Integers

I have a question as such: A set $\{a_1, \ldots , a_n \}$ of positive integers is nice iff there are no non-trivial (i.e. those in which at least one component is different from $0$) solutions ...
1answer
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### Show a subring contains certain elements.

Show that the set of all real numbers of the form $a_0 + a_1\pi + a_2\pi^2 +\cdots+ a_n\pi^n$ with $n≥0$ and $a_i ∈ \mathbb{Z}$ is a subring of $R$ that contains $\mathbb{Z}$ and $\pi$. ...
1answer
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### How can I prove that this function is uniformly continuous?

How to show that $f(x)=\frac{x}{1+|x|}$ is uniformly continuous? Thank you. Also, how do I become good at writing these proofs?
1answer
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### Proving if the limit of f(x) approaches zero, then the limit of 1/|f(x)| approaches infinity.

The Problem: Suppose that $f:D\to\Bbb R$, where $D$ is a subset of $\Bbb R$ and $a$ is an accumulation point of $D$, $\lim_{x\to a}f(x)=0$, and $f(x)\ne0$ for any $x$ in $D$ in some neighborhood of ...
1answer
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### For bijection $f:A \rightarrow B$, prove that $f^{-1} \circ f = {\text {id}}_{A}$

I have to prove that for a bijection $f:A \rightarrow B$, $f^{-1} \circ f = {\text {id}}_{A}$, where ${\text {id}_A}$ is the identity function of $A$, and we define $f^{-1}: B \rightarrow A$ by ...
0answers
79 views

### Proving direct sums.

Let $V$ be a vector space over $F$ and let $U$ and $W$ be subspaces such that $V=U+W.$ Prove that $V=U\oplus W$ if and only if $U\cap W=\lbrace 0 \rbrace$. Attemp: Suppose $V=U\oplus W$. By ...
1answer
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### Contradiction Proof regarding Well-Ordering Principle

Let $r_0$ be the smallest element of a set $S$ such that $S\subseteq\mathbb {N} \cup \{ 0 \}$. According to the Well-Ordering Principle, this implies that $r_0$ $\ge 0$ and $r_0 = a - q_0 b$ for some ...