# Tagged Questions

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### How do I turn my verbal argument into something formal in [Real Analysis]? (proving every compact set is bounded)

So one of the exercises I am doing is to prove (or disprove) that 'Every compact set on a metric space is bounded'. Verbally, I can 'prove' this by simply stating: "If the every compact set on a ...
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### How do I turn my verbal argument into something formal in Real Analysis? [duplicate]

So one of the exercises I am doing is to prove (or disprove) that 'Every compact set on a metric space is bounded'. Verbally, I can 'prove' this by simply stating: "If the every compact set on a ...
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### How to finish this proof about compact implies bounded

A set is called compact if every sequence has a convergent subsequence. I am trying to show: If $K$ is compact then it is bounded. (that it is closed was very easy to prove) What I want to do: Let ...
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### Lipschitz continuous set-valued map

Let $X$ and $Y$ be metric spaces and $F:X\to2^Y$ be a set-valued map. Suppose that $2^Y$ is endowed with the Hausdorff metric. I wonder about sufficient conditions on $F$ that ensure this map is ...
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### We know that a Compact set is closed. However a finite discreet set is compact but not closed (contradicting the theorem?)

We know that a Compact set is closed. We also know that a finite discreet set is compact (as every cover has a finite sub cover). However a finite discreet set is not closed (contradicting the ...
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### metric-spaces closure

The line in my writing symbolizes the closure, where we add all the boundary points. Is it true that: $\overline{A \cap B}=\overline{A}\cap {B}$? If B is closed, but we do not know if or if not A ...
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### Understanding a statement about equivalent norms ($||\cdot ||_2 \sim||\cdot||_1$)

I am trying to understand the following statement from an analysis book: Two norms are equivalent ($||\cdot ||_2 \sim||\cdot||_1$) if they induce equivalent metrics. At first I thought this ...
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### Proof verification related to the discrete metric

Can someone please verify my proof? Let $X_1$ be a set and let $d_1$ be the discrete metric on $X_1$. (a) Prove that every subset of $(X_1, d_1)$ is open. (b) Prove that if $(X_2, d_2)$ ...
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### Condition for Lipschitz functions and set inclusion

Let $(X,d)$ be a pseudometric space, and for each $A\subseteq X$ and $\varepsilon \geq 0$ define $$A^\varepsilon := \{x\in X:\exists a\in A \text{ s.t. }d(x,a)\leq \varepsilon\}.$$ Is this set ...
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### Qualifier problem: Completeness of Metric Spaces

I am working on old qualifier problems as a review, and I came across this one: Suppose there exists a continuous surjection $f:X_1 \mapsto X_2$, where $(X_1,d_1),(X_2,d_2)$ are metric spaces, such ...
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### if $A, B$ are open in $\mathbb R$ then so is $A+B.$

I am trying to find out a counterexample to the problem: if $A, B$ are open in $\mathbb R$ then so is $A+B.$ But I could not find any such counterexample. Please help me.
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### How to check whether this function is continuous or not..?

Let A and B be two disjoint closed sets of any Metric space X.I have to construct a continuous function such that $f(x):= 0$ if x belongs to $A$ $f(x) = 1$ if x belongs to $B$ My idea is to use the ...
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### Characterization of continuity by subsequences.

I have a difficulty trying to prove the following proposition. Any help would be greatly appreciated. $\textbf{Prop.}$ Let $(X,d_1)$ and $(Y,d_2)$ be two metric spaces. A function $f:X\rightarrow Y$ ...
I read that the formula $d(x,y)=\mathrm{arccosh}(1+\frac{2||x-y||^{2}}{(1-||x||^{2})(1-||y||^{2})})$, where $x,y$ are in the open unit ball of $\mathbb{R}^{n}$ and $||\cdot||$ denotes Euclidean norm, ...