# Tagged Questions

1answer
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### Proving a sequence is increasing and converging as $n\to \infty$.

Suppose that $x_0 \in (-1,0)$ and $x_n=\sqrt{x_{n-1}+1}-1$ for $n \in \mathbb N$. Prove that $x_n \uparrow 0$ as $n\to \infty$. What happens when $x_0 \in [-1,0]$? Before this, the problems I did had ...
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4answers
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### Lowest possible value of a function with derivative greater than 2

I have the following two problems, and want to attempt to solve them with Mathematical rigour(which I don't yet possess): Suppose that $f$ is differentiable on $[1,4]$ and is such that $f(1) = 10$ ...
0answers
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### U-substitution proof by partitions

I am trying to prove integration by substitution, i.e. for $f:[c,d] \rightarrow \mathbb{R}$ continuous and $\phi: [c,d] \rightarrow [a,b]$ continuous on $[c,d]$ and $C^1$ on $(c,d)$. Then define ...
2answers
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### Show a double-sided infinite integral of $\sin(x+b)$ exists iff $b=n\pi$

More formally: Show that $$\lim_{a\rightarrow \infty} \int_{-a}^a \sin(x+b)$$ exists if and only if $b=n\pi$ for some $n \in \mathbb{Z}$. I get the intuition fine. The function is just a horizontal ...
0answers
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### Proving u-substitution the hard way — use only definition of integration with partitions

I am trying to prove integration by substitution, i.e. for $f:[c,d] \rightarrow \mathbb{R}$ continuous and $\phi: [c,d] \rightarrow [a,b]$ continuous on $[c,d]$ and $C^1$ on $(c,d)$. Then define ...
3answers
104 views

### Existence of increasing sequence $\{x_n\}\subset S$ with $\lim_{n\to\infty}x_n=\sup S$

Let $S\subset\mathbb{R}$ be a non-empty bounded above set. Then there exists a monotone increasing sequence $\{x_n\}\subset S$ such that $$\lim_{n\to\infty}x_n=\sup S.$$ I'm struggling with ...
0answers
36 views

### Proof Validation Function From Integers to Rationals is Continuous

I am teaching myself real analysis, so any help is greatly appreciated. Let the function be defined as $F : Z \rightarrow Q$ where $Z$ is the set of integers and $Q$ is the set of rational numbers, ...
1answer
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1answer
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### When proving things how do I defer choosing a values?

The best example is what I've just tried to prove. Usually I do these proofs in 2 or 3 passes, or draw a margin to separate notes. In the example I want to defer picking an $\epsilon_f$ and ...
0answers
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### Show that this is not differentiable at any point in $\mathbb{R}$

Define $\phi: \ \mathbb{R} \rightarrow \mathbb{R}$ by $$\phi(x) = \begin{cases} x\ :\ 0\le x\le \frac{1}{2}\\ 1-x :\ \frac{1}{2} \le x \le 1 \end{cases}$$ And then extend ...
1answer
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### Prove the least upper bound property using Bolzano Weierstrass theorem

Prove the least upper bound property using Bolzano Weierstrass theorem. I know there are quite a fair number of similar questions on the site, but none of them provide satisfactory proofs. Does ...
1answer
201 views

### Proof concerning equivalent definition of supremum using limits

I believe I have a proof to the following theorem: Let $S$ be a subset of $\mathbb{R}$ that is non-empty and bounded above. $s \in \mathbb{R}$ is the supremum iff $s$ is an upper bound of $S$ and for ...
0answers
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### Let {$p_n$} be a sequence of points in the $\mathbb{R}^2$. Use the notion of convergence to solve the following

A) Define what it means for a point p $\in$ $\mathbb{R}^2$ to be a limit point of {$p_n$}. B) Prove that p is a limit point of {$p_n$} if and only if {$p_n$} has a subsequence which converges to p. ...
0answers
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### My first simple direct proof (very simple theorem on real numbers). Please mark/grade.

What do you think about my first simple direct proof? What mark/grade would you give me? Besides, I am curious about whether you like the style. Theorem Let $I = [a,b]$ be a non-empty closed ...
1answer
40 views