What is wrong with this proof? (Induction)?

it's entirely circular, that's what's wrong with it. it's like saying "all plants are round and green. i will prove this because peas are round and green. now suppose this applies to all plants. then all plants would be round and green. see if you can figure out what is wrong with this proof." you're not supporting the induction with anything and i'm not really sure why you're asking this.

I think I get the question now. Let P(n) be the proposition that all people in a group of n have the same name. As part of the inductive process, you must show that P(n) must be true if P(k) is true for all k < n.

The trouble begins when n = 2. Suppose that the two people are Bob and Doris. First we remove Bob. All of the remaining people have the same name, because P(1) has been proved (or accepted as obvious anyway). Now replace Bob and remove Doris. Again, all of the remaining people have the same name. Here is the mistake:

"...they all have the same name, which shows that the person we excluded originally has the same name as all the others."

No, it does not prove that. The person we originally excluded (Bob) was never compared with Doris directly. And when n = 2, we cannot use any kind of a transitive property, because there are no people with whom Bob and Doris have both been compared.

The proof is invalid, because you already ASSUMED the entire group has the same name.

If the group already has the same name, it is trivial to prove a subset has the same name!

The problem is you need to first prove it true for ANY n, not just n = 1.

Induction usually proves for some value, n=1, and then proves it true for n+1, NOT n-1, as you did.

no longer an entire answer...yet i'm positive i can help you you if I understand.... i do not see how your evidence for ok+a million is valid. You tell me that in case you eliminate one huge style from a chain of ok equivalent numbers that the perfect numbers are all a similar. that is clearly real. in spite of the indisputable fact that this shows me it is your assumption is real for a chain of ok numbers then it really is likewise real for ok-a million individuals of that set. It does no longer instruct me that it really is real in case you upload an more desirable huge style ...it really is for a chain of ok+a million numbers.... Please clarify why you've self assurance that your evidence demonstrates this. ok thanks for the further clarification...yet...i'm nonetheless no longer confident, the evidence would not fairly make experience to me....something is incorrect right here. obviously if this works for a 'universal' set the position all numbers are a similar..then it also ought to paintings for a particular set. So why no longer instruct me your evidence the position we start up with a chain of ok numbers...that are all equivalent to x, and let ok be 5 and x b 12. So, in different words you've a chain of 5 numbers all equivalent to 12....Please use your good judgment to instruct me that in case you upload a sixth huge style to the set that this huge style also must be 12..... obviously this isn't the case, yet my wager is that operating by this get jointly will instruct you the mistakes contained in the best judgment....Please try this as i might want to need to understand the "concern" more desirable suited...(excellent now i do not even see an situation as to me the evidence for ok+a million isn't maximum suited) i'm satisfied to debate more desirable if this facilitates.

The problem with this is that it is not mathematics, and therefore not mathematical induction. Try the section dealing with Logic.

"now suppose that the theorem is true for all groups of people of size k < n" this just doesn't make sense.

induction is two steps:you first prove statement true for n=1 then you prove that if it is true for any n it must be true for n+1