# Tagged Questions

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### Proving properties of closures using intersection of indexed sets and topology

How would I write a proof for this example? Let X be a set, and let $B \subseteq \mathcal P \left({X}\right)$. Define $B^* =${ $U \subseteq X:$ There is an index set $I$ and $U_{i} \in B$ for each ...
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### Help to clarify inductive step for proof of $\mathbb{N_m} \rightarrow \mathbb{N}_n\Rightarrow m\le n$

The statement to prove is: If there exists an injection $\mathbb{N_m} \rightarrow \mathbb{N}_n$ then $m\le n$ The solution says to prove by induction on $n$. I just need help on the inductive ...
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### Prove that $n<(3/2)^n$ for any $n$ with induction [closed]

need help with induction with inequality, I suck at it. $n<\left(\frac{3}{2}\right)^n$ for any $n$
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### Question on Induction (Very Simple)

I've just started a course in mathematics at university, and our current topic is mathematical induction. I've been given the following question: $$1+4+4^2+....+4^{n-1}=\frac{4^{n}-1}{3}.$$ I get ...
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### In proof by induction, what does it mean when condition for inductive step is lesser than the propsition itself?

My question is regarding the question posed at the end of the proof. My answer is that the result does not hold for all $m \ge 7$ because when $m=7$, the result is $343 \le 128$, which is false. ...
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### Base case in the Binet formula (Proof by strong induction)

I don't understand why the author says that the inductive formula does not apply until $u_3$. The formula does not depend on other terms in the Fibonacci sequence, so I thought I could just prove it ...
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### My first proof that uses the well-ordering principle (very simple number theory). Please mark/grade.

What do you think about my first proof that uses the well-ordering principle? Please mark/grade. Theorem The sum of the cubes of three consecutive natural numbers is a multiple of 9. Proof ...
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### proof by induction: sum of binomial coefficients

Question: Prove that the sum of the binomial coefficients for the nth power of $(x + y)$ is $2^n$. i.e. the sum of the numbers in the $(n + 1)^{st}$ row of Pascal’s Triangle is $2^n$ i.e. prove ...
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### My first proof employing strong induction / complete induction (very simple number theory). Please mark/grade.

What do you think about my first proof employing strong induction? What mark/grade would you give me? Theorem Every natural number greater than 1 is a product of one or more primes. Proof First, ...
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### My first induction proof (very simple number theory). Please mark/grade.

What do you think about my first induction proof? Please mark/grade. Theorem The sum of the cubes of three consecutive natural numbers is a multiple of 9. Proof First, introducing a predicate ...