-2
votes
2answers
59 views

What's wrong with this induction based proof?

Claim: $\forall x \in \mathbb{R^+} ,$ $ x^n=1 $ $where$ $ n\in \mathbb{N}$ Proof by induction on n: Basis step: $\forall x \in \mathbb{R^+} ,$ $ x^0=1 $ Induction Step: Let this holds for all ...
2
votes
0answers
50 views

Simple knapsack with arbitrary weights: Algorithm won't work, but my proof by induction doesn't agree.

We want to solve the simple knapsack problem: We're given a set of $n$ positive item weights, which are unique integers $\{w_1, \ldots , w_n\}$, and an integer $C > 0$, representing the capacity of ...
3
votes
2answers
237 views

Fallacious Induction Proof that All Integers are Equal [duplicate]

We're currently learning about induction in my real analysis course. I came up with the following proof that is obviously false but cannot quite figure out why it fails. Here it is... "Any set $S$ of ...
1
vote
1answer
79 views

Am I correctly identifying the fallacy in this induction “proof?”

The prompt states: Let us accept as true that a person can always walk an extra mile. Does the Principle of Induction then prove that a person can walk forever? Where is the fallacy? No. ...
2
votes
1answer
106 views

What is wrong with this induction?

Let $P(n)$ be any property pertaining to a natural number $n$. We will look this example: $$P(n) := (n = 0) \vee (n \leq -1) $$ Now, I will prove this and I'm asking that can you please show me where ...
0
votes
2answers
469 views

Flawed proof that all positive integers are equal

Suppose that we are trying to prove that for every positive integer n, if x and y are positive integers with max(x, y) = n, then x = y. For the base case, we suppose n = 1. If max(x, y) = 1 and x and ...
16
votes
8answers
2k views

There are no bearded men in the world - What goes wrong in this proof?

Several years ago in a textbook I read this example as a faulty use of proof by induction. I never really realized why it fails. Here it goes: Theorem. There are no bearded men in the world. ...
47
votes
12answers
4k views

All natural numbers are equal.

I saw the following "theorem" and its "proof". I can't explain well why the argument is wrong. Could you give me clear explanation so that kids can understand. Theorem: All natural numbers are ...
2
votes
2answers
347 views

Find the demonstration error for the statement “All positive integers are equal”

All positive integers are equal, that is, for each $n \in \mathbb{N}$ the assertion $P(N): 1 = \cdots = n$ is true. (i) $P(1)$ is true because $1 = 1$ (ii) Suppose that $P(n)$ is true, then $1 = ...
3
votes
1answer
110 views

Find the fallacy in the following treatment

Claim: any two positive integers are equal Proof: Let $A(n)$ be statement: if $a$ and $b$ are any two positive integers such that $\max(a,b)=n$ then $a=b$ Suppose $A(r)$ is true. Let ...
4
votes
1answer
524 views

What’s bogus about this Strong Induction Proof on weakly decreasing sequence of primes?

I couldn't find what is wrong with this strong induction proof, any one knows ? Question: A sequence of numbers is weakly decreasing when each number in the sequence is $\geq$ the numbers after it. ...
5
votes
3answers
646 views

Fake induction proof

Using the induction method: $(\forall P)[[P(0) \land ( \forall k \in \mathbb{N}) (P(k) \Rightarrow P(k+1))] \Rightarrow ( \forall n \in \mathbb{N} ) [ P(n) ]]$ Why this proof is wrong? $P(x)\equiv ...
43
votes
2answers
2k views

Proof of 1 = 0 by Mathematical Induction on Limits?

I got stuck with a problem that pop up in my mind while learning limits. I am still a high school student. Define $P(m)$ to be the statement: $\quad ...