# Tagged Questions

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### Is this a metric on R?

For $x,y \in \mathbb{R}$, define $d(x,y) = \sqrt{|x-y|}$. Is this a metric on $\mathbb{R}$? It's clear that $d(x,x) = 0$ and $d(x,y) = d(y,x)$ for all $x,y \in \mathbb{R}$. The triangle inequality ...
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### Sequence Criterion of Continuity and Divergent sequences

A function $f : (X, d_X) \to (Y, d_Y)$ between metric spaces is continuous iff $$\lim_{n \to \infty} f(x_n) = f(\lim_{n\to \infty} x_n)$$ for each convergent sequence $(x_n)$ in $(X, d_X)$. So if I ...
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### Show the following is Cauchy:

I am trying to prove that the Euclidean Norm/inner product on C([0,1]) does not give rise to a complete metric space. To do this I am trying to find a Cauchy Sequence which does not converge in ...
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### Is this claim harder to prove for arbitrary metric spaces than for the reals?

Let $X$ and $Y$ be metric spaces. Define the distance between functions $f, g$ from $X$ to $Y$ as $$d(f, g) = \sup_{x \in X} \frac{d(f(x), g(x))}{1+d(f(x), g(x))}$$ Is it true that if $f_n:X \to Y$ ...
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### Prove that if sets $A$ and $B$ are closed and bounded then $A+B$ is closed

Prove that if sets $A$ and $B$ are closed and bounded then $A+B$ is closed I know that $A$ and $B$ are closed and bounded, then they are sequentially compact, so $A+B$ also sequentially compact, ...
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### Using the hypothesis that an equicontinuous sequence is pointwise bounded to show it's uniformly bounded

I'm making steady progress in this problem, but it is a little unclear on where exactly to use the fact that $\{f_n\}$ is pointwise bounded. I was going the route of a compactness argument, but I'm ...
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### Set that is closed and bonded, but not compact?

Let $\mathbb Q$ be the set of rational number with d(p,q) = |p-q| and E be the set of all p $\in \mathbb Q$ such that $2 < p^2 < 3$. Intutively, I think about closedness using subspace ...
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### Showing $\rho (x,y)=\frac{d(x,y)}{1+d(x,y)}$ is a metric space.

Show $\rho (x,y)=\dfrac{d(x,y)}{1+d(x,y)}$ is a metric on the metric space $X$, equipped with the Euclidean metric $d$. I've already shown that the positivity $\rho(x,y)\geq 0$, the symmetry ...
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### Metric spaces, compactness, how to solve this with the open covering property?

I have an excercise I am supposed to solve with the open covering property. I am not able to do that, but I can solve it with another method. I have two questions. 1. Is my method of solving it ...
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### When is $\mathcal C(X)$ complete?

I learned today in class that $\mathcal C([a,b])$ is complete with the supremum norm. That is, any uniformly convergent sequence $(f_n)$ is Cauchy, and the converse is true. I asked my teacher about ...
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### Concerning $C^0[0,1]$ and the $L^1$-Norm.

Consider the well known Euler sequence of functions $x^n$ ($n=1,2,3\ldots$) on $[0,1]$. It is clear that it converges against $\chi_{1}$, the characteristic function of the singleton $1$, in the ...
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### How to prove that this set is open.

This is probably very easy. I know that it is very obvoius, but I want to prove it using the definition of what beeing open means. I have $A = \{y \in R: y <a \}$. And I want to show that it is ...
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### On proving that a compact metric space is bounded

I am trying to prove the above claim. My question is on the bolded part and if I am correct in making that statement. A metric space is bounded if there exists an $M \in \mathbf{N}$ such that ...
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### Given data, approximations in a metric space for moving into a normed vector space isometrically.

Please see this question and this answer. Here $f_x(y)$ is approximated by $$x_v = [d(x,K_1),d(x,K_2),....d(x,K_N)]$$ by choosing to consider distances from $x$ to only certain points $K_i$ and ...
### Metric space $X=(0,1)$ with $d(x,y):= \vert \tan x-\tan y \vert$ complete or not? [duplicate]
Let $X=(0,1)$ be a metric space with metric defined as $d(x,y):= \vert \tan x-\tan y \vert$ for all $x,y \in (0,1)$. The question is whether $X$ is complete with respect to the given metric or not. ...