what is wrong with this proof using induction

[proven swallow]

I think the problem is at (2). Just because a1=a2=...=ak doesn't imply ak=a(k+1). You cannot apply the hypothesis for (2) because it only supposes that a1 to ak are equal and does not involve a(k+1) in any way.

For example, let's say P(4) is true. Then a1=a2=a3=a4. Does this mean a4=a5? No, because I can still choose a5 to be whatever I want.

[feral anvil]

I dont think that is the problem

equality is transitive so you can set all the ak to be equal

The problem is that the base case only has 1

But thats not a sufficient base case

Your base case here needs to be P(2) as well

which means youd have to show a1=a2 for any two real numbers a1 and a2

which u obviously cant so the proof falls apart here

When ur doing the inductive step

youre making the implicit assumption that any two values are equal

i.e k>= 2

So that you actually have something like a1=a2

AND then a2=a3

so then you can use transitivity to show a1=a3

but the base case here only does it for one number

This is like the 'all horses are black' example for a wrong inductive proof

[glossy delta]

i know this proof told a bit differently

i know this as:
prove by induction that the color of every horse in the world is the same

so yes

the problem is in 2

because its like removing one object from the current group and adding 2

and we can't take away an object from a 1 number group cuz then we are left with... no numbers

but it is intuitive to think about with horses

and also intuitive to see why problem is in 2

because if its true that every 2 horses are the same colors

so we can make a group of 3 object equal with transition

a=b, b=c => c=a

so from the 2 horses n

we can say that 1st horse and 2nd horse are same color

and 2nd horse, 3rd are same color

thats why 3 and 1 are same

but the problem is that this induction step doesn't work from 1 to 2