For questions about the formulation of a proof, not about the mathematics behind it.

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1answer
25 views

How can we fill in some missing details in this proof?

Let $(X,d)$ and $(Y,\rho)$ be metric spaces and $X$ is compact. Suppose $f:X\to Y$ and $f_n:X\to Y$ are continuous functions such that for every $x\in X$, $\rho(f_n(x),f(x))$ decreases to $0$ as $n\...
0
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2answers
169 views

Suppose that $A$ and $B$ are subsets of $X$ such that $A \subseteq B$ then $Int(A) \subseteq Int(B)$.

Let $(X, \mathfrak T)$ be a topological space and suppose that $A$ and $B$ are subsets of $X$ such that $A \subseteq B$ then $Int(A) \subseteq Int(B)$. I know this a true statement so now I need to ...
1
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3answers
98 views

Book Recommendations for Writing Proofs

As an applied mathematics student coming from a small university, I have not had an adequate course in writing/formulating proofs for problems in advanced calculus/real analysis (my university has an ...
1
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1answer
39 views

Formulating a problem in terms of set theory

Here is one problem I was trying to solve just by trial-and-error method. However, I was thinking about how to write the clear solution using set theory. Problem: A notebook contains exactly $100$...
0
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2answers
35 views

binomial inequality with sums

Assume I have a series of numbers $a_1 \dots a_n$ where $0 \leq a_i \leq n-1$ and a positive integer $r$. how to show that the sum of number of ways to choose $r$ from $a_i$ is at least as $n$ times ...
2
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4answers
6k views

Proof by induction that $ \sum_{i=1}^n 3i-2 = \frac{n(3n-1)}{2} $

I'm starting to understand how induction works (with the whole $k \to k+1$ thing), but I'm not exactly sure how summations play a role. I'm a bit confused by this question specifically: $$ \sum_{i=1}^...
1
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2answers
43 views

Could someone please check my proof that $(x_n) \text{ is Cauchy in } X\implies (f(x_n)) \text{ is Cauchy in } Y$

Let $(X,d)$ and $(Y,\rho)$ be metric spaces and $f:X\to Y$. Show that if $f:X\to Y$ is uniformly continuous, then $$(x_n) \text{ is Cauchy in } X\implies (f(x_n)) \text{ is Cauchy in } Y$$ My ...
0
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0answers
30 views

How to get a feel for rigor/form used in mathematics?

I'm an engineer, and while you get introduced to many concepts of mathematics, but only with a subset of the vocabulary, and none of the rigor and proofs. So while trying to read a mathematical book, ...
1
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1answer
29 views

Is it sufficient to prove that a function is an open map by looking at the basis element?

I am trying to prove that the projection map $\pi_X:(X, T)\times (Y,J) \to X$ is an open map But I don't know if I can use the basis element directly, so my proof is quite round about and lengthy ...
0
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1answer
849 views

Disjunctive normal form (BOTH dnf and cnf) example help

T or ( not x and y ) or ( x and y ) In class we went over examples of how many things can be both DNF and CNF... eg. (not z) or y can be thought of as both (not z) or y .... dnf ((not z) or y)...
1
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1answer
15 views

Show that there exists at most one extension of $f$ whose co-domain is a Hausdorff space [duplicate]

I want to show the following Suppose $A \subset X, f: A \to Y$ is continuous, $Y$ is Hausdorff. Show that there is at most one continuous extension $g: \overline A \to Y$ I feel like I am ...
1
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2answers
44 views

How to see that $f(t) = (t, 2t, 3t, \ldots)$ continuous in the product topology

I am trying to check whether $f: \mathbb{R} \to \mathbb{R}^\omega$ $f(t) = (t, 2t, 3t, \ldots)$ is continuous or not in the product and box topology. But I have a feeling I don't have the ...
3
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3answers
63 views

Topology: is it ever good to write $x \in U \in \mathfrak{T}$

Sometimes I come across a sentence in my topology book that says, let $U$ be an open set that contains $x$ I can't help but write it down as: Let $$x \in U \in \mathfrak{T}$$ Is it good ...
0
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0answers
19 views

Fisher information for exponential family: Regularity conditions

for the Fisher-Information to be defined certain regularity conditions have to be fulfilled (like in Lemma 5.3. in Theory of Point Estimation by E.L. Lehmann or on slide 2 here: http://www.stat.nus....
3
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4answers
93 views

$a^n$ even implies $a$ even

I've tried to prove that $(\forall a,n>0 \in \mathbb{N}),(a^n \text{ even} \implies a \text{ even})$, can someone tell me whether my proof is sound? Lemma 1: $a \text{ even} \implies a^2 \text{ ...
2
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0answers
47 views

proof of Triangle Removal Lemma

Where can I find a proof of the following version of Triangle Removal Lemma (or any version equal to it)? Let $G(V,E)$ be a graph on $n$ vertices such that it contains $\varepsilon n^3$ triangles, ...
0
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2answers
51 views

Prove that the directrix-focus and focus-focus definitions are equivalent

(NOTE: This is my attempt at answering this question and this question, but I rewrote it in order to make it easier for me to solve. Also, I've made a YouTube video explaining this whole proof. Note ...
0
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1answer
13 views

If unions of two families sets are disjoint then families of sets are disjoint too.

I have read that theorem "Suppose $\mathcal{F}$ and $\mathcal{G}$ are families of sets. If $\cup\mathcal{F}$ and $\cup\mathcal{G}$ are disjoint, the so are $\mathcal{F}$ and $\mathcal{G}$" is ...
3
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1answer
119 views

If $g$ is Riemann-integrable in a closed interval and $f$ is a increasing function in a closed interval, is $g\circ f$ Riemann-integrable?

If $g$ is Riemann-integrable in a closed interval and $f$ is a increasing function in a closed interval, is $g\circ f$ Riemann-integrable? To clarify: the problem stated that the composition is well ...
2
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2answers
43 views

Proof verificication and question of rigour: $A$, $B$, connected implies union is connected

Don't mark this as duplicate. The other question does not help me figure out how rigorous MY proof is. Problem: Let $A$ and $B$ be connected subsets of a metric space and let $A\cap\overline{B}\neq\...
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votes
3answers
73 views

prove that the $5$th root of $r$ is irrational if $r$ is irrational [on hold]

I am trying to learn mathematics for computer science in own efforts. I got this task to prove that $\sqrt[5]{r}$ is irrational, given that $r$ is irrational. Normally if I had to prove that $\sqrt{2}...
0
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0answers
63 views

Analysis, Groups and a very elementary Proof [on hold]

I am studying Groups at early undergrad level and am very interested in analytical proofs such as: $$For\;any\; x \in R,\; x.0=0$$ $$Proof\qquad By\;A3,\; 0+0 = 0\;and\;so\;x.(0+0)=x.0.\;Hence,\;by\;...
0
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1answer
42 views

Nice way to prove a limit.

I know how to prove the following limit $$\lim _{\epsilon \rightarrow 0} \frac{a^{\epsilon}-1}{\epsilon}=\ln(a)$$ But I am looking for a nice way to do it, a little elegant. Would you have one?
1
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3answers
56 views

Prove $\forall n \in \mathbb{N}$ where $ n \neq 1 , n + \frac{1}{n} > 2$ using completing the square.

I have got this far; I am only unable to understand how to finish the proof. $n>0 \implies n + 1/n > 0 \implies n + 1/n + 2 - 2 > 0 \implies {\big(\sqrt{n}+\frac{1}{\sqrt{n}}\big)}^2 - 2 >...
4
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1answer
1k views

If $\lim f(x) = 0,$ then $\lim 1/|f(x)| = \infty.$

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 ...
30
votes
9answers
3k views

Why don't Venn diagrams count as formal proofs?

Just curious. If the purpose of a proof is to inform and persuade, why don't Venn diagrams count? Is it just convention or is there a more, umm, formal reason haha. Thanks!
2
votes
8answers
200 views

Prove that $4$ divides $3^{2m+1} - 3$

Prove that $4$ divides $3^{2m+1} - 3$. By plugging in numbers I can see this is true, but I can't figure out a way to prove this, I was thinking maybe proving first that it is divisible by $2$, and ...
1
vote
2answers
29 views

Show that a set is open if and only if each point in S is an interior point.

I am in a complex analysis class and have been asked to prove this. I know I have to prove both ways so. If a set is open then each point in $S$ is an interior point. Proof: Let $S$ be an open set,...
2
votes
6answers
100 views

Three positive numbers a, b, c satisfy $a^2 + b^2 = c^2$; is it necessarily true that there exists a right triangle with side lengths a,b and c?

If so, how could you go about constructing it? If not, why not? I am new to proofs and I was reading a book where they posed this question. I understand that if we are given any right triangle, the ...
2
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1answer
96 views

How can I prove $\sqrt{2} ^{\sqrt{2}}$ is irrational? [duplicate]

I am learning proofs and a question was posed which asked us to prove that $\sqrt{2}^{\sqrt{2}}$ is irrational. They mentioned this - Hint: try using the log10 function... I tried my hand at the ...
4
votes
2answers
88 views

problem proving: $(1+q)(1+q^2)(1+q^4)…(1+q^{{2}^{n}}) = \frac{1-q^{{2}^{n+1}}}{1-q}$

I'm trying to prove this, and it is really frustrating, because it seems a really easy problem to prove, however, I'm having a little problem with exponents: $$(1+q)(1+q^2)(1+q^4)...(1+q^{{2}^{n}}) = ...
0
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1answer
58 views

Proof that $|S| \leq |T|$ if $S \subseteq T$.

Let $S$ and $T$ be sets. I am having trouble proving that if $S \subseteq T$, then $|S| \leq |T|$, where $|S|$ is the cardinality of $S$.
2
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3answers
38 views

Principle of Mathematical induction proof

Prove that $2^n >n$ for all positive integer $n.$ I know this can be easily proved by using PMI Let $P(n): 2^n > n$ For $n = 1$ $$2^1 > 1.$$ Hence $P(1)$ is true. Assuming that $P(k)$ is ...
1
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6answers
162 views

Proving that $64$ divides $3^{2n+2}+56n+55$ by induction

Let $n ≥ 0$ be an integer. Prove by induction: 64 divides $3^{2n+2} + 56n + 55$ general expression: $3^{2n+2} + 56n + 55 = 64m$ 1st I substitute $P(0)$ and it gives me true: $9+55 = 64$ (if m = 1 ...
0
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2answers
46 views

(Real Analysis) Topology: Prove $f(cl S)\subseteq clf(S)$

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be continuous. Show: $f(\overline{S})\subseteq \overline{f(S)}$ for $S\subseteq \mathbb{R}$ (Note: $\overline{S}$ denotes the closure of S; $\partial S$ ...
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0answers
15 views

there exists a gray code of length 2k for any positive integer k [closed]

Can any one help me prove the statement "there exists a gray code of length 2k for any positive integer k" using mathematical induction thanks
3
votes
1answer
779 views

Counting Elements and Their Inverses

The problem I am attempting to prove is the following: In any finite group $G$, the number of elements not equal to their own inverse is an even number. Caveat: I have had very limited experience ...
4
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1answer
73 views

Theorem 2.27 (a) in Baby Rudin: Is his proof complete enough?

Here's Theorem 2.27 (a) in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: If $X$ is a metric space and $E \subset X$, then $\overline{E}$ is closed. Now here's ...
0
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2answers
52 views

(Rigor/Validity of Proof) Every sequence of reals in a compact set has a convergent subsequence

[ADDED/MODIFIED]: I began my proof with a compact set, but this was a wrong start. Although the comments are valid, I should've started with a bounded set. Because what I want to establish first is ...
118
votes
9answers
8k views

Why do people use “it is easy to prove”?

Math is not generally what I am doing, but I have to read some literature and articles in dynamic systems and complexity theory. What I noticed is that authors tend to use (quite frequently) the ...
4
votes
6answers
118 views

Prove that $ 1+2q+3q^2+…+nq^{n-1} = \frac{1-(n+1)q^n+nq^{n+1}}{(1-q)^2} $

Prove: $$ 1+2q+3q^2+...+nq^{n-1} = \frac{1-(n+1)q^n+nq^{n+1}}{(1-q)^2} $$ Hypothesis: $$ F(x) = 1+2q+3q^2+...+xq^{x-1} = \frac{1-(x+1)q^x+xq^{x+1}}{(1-q)^2} $$ Proof: $$ P1 | F(x) = \frac{1-(...
0
votes
2answers
20 views

Strong Induction issue

I am trying to prove a statement using strong induction but I seem to be getting stuck. I don't know if did something wrong or I am just not recognizing an opportunity for factoring/how to factor ...
1
vote
2answers
65 views

Proof that $\overline{P(z)} = P(\overline{z})$ for polynomial $P$ with real coefficients

Let $$ a_0, a_1, a_2, a_3, \ldots , a_n \quad (n \ge 1)$$ denote real numbers, and let $z$ be any complex number. With the aid of $$ \overline {z_1 +z_2+ \ldots +z_n} = \overline z_1 +\overline z_2+ \...
0
votes
2answers
45 views

Easy proof the set of finite Set in countable is countable [duplicate]

Suppose I know a result that the set of finite sets in $\mathbb{N}$ is countable. Is there a very quick way to show that the set of finite sets in any $X$ countable is countable? Idea...two sets ...
0
votes
1answer
19 views

Proving $g(x) = E_x$.

I'm pretty new to mathematical proof and set theory and I'm having trouble proving this problem. I've started on it, but I don't know where to progress. Problem: Suppose that $f: A \rightarrow B$. ...
2
votes
2answers
40 views

Topology: Show restriction of continuous function is continuous, and restriction of a homeomorphism is a homeomorphism

I need to prove two trivial results but I don't know how to work with restricted function and its inverse Consider the topological spaces $(X, \mathcal{T}), (Y, \mathcal{J})$ Claim 1: Let $f:X \...
0
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1answer
28 views

Show that every local homeomorphism is continuous and open therefore bijective local homeomorphism is a homeomorphism

Follow up on another question I asked recently: Topology: Show restriction of continuous function is continuous, and restriction of a homeomorphism is a homeomorphism Definition: Let $(X, \mathcal{...
0
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2answers
46 views

Show that cuts are preserved under homeomorphism

Let $(X, \mathcal{T})$ be a topological space, assume that $X$ has no proper (not $X$ or $\varnothing$) clopen subset. Definition: A point $p \in X$ is a cut if $X \setminus\{p\}$ has a proper ...
-1
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1answer
42 views

Show that a space is $T_1$ if all subset is the intersection of all open sets containing it

Following a question I asked yesterday Is this the correct definiton of $T_1$ space? I was left with a claim: $(X, \tau)$ is $T_1$ iff $\forall A \subseteq X, A = \bigcap\{U \subseteq X: U \in \...