Question

Prove \[\forall a \in \mathbb{R}, a * 0 = 0\]

Answer 1

use the fact that 0 + 0 = 0

Answer 2

you are trying to prove that for all real constants a, a*0 = 0
cant you just say that you multiply a*b you add b a tmes. thus a*0 is 0+0+0+0 a times and as 0+0=0 a*0 is also 0?

Answer 3

or is not mathematical enough

Answer 4

that's what I was trying to prove: a*0 = 0. your proof assumes that it is already true, when I only know for a fact that 0 + 0 = 0, but I don't know yet that a*0 = 0

Answer 5

sumbul to the rescue :-D

Answer 6

consider ax0=ax(1+(-1)) (by existence of additve inverse)
=aX(1)+aX(-1) by distributive law
=a+a(-1) by existence of multiplicative identity
=a+(-a) (because a(-1)=-a)
=0 by existence of additve identity

Answer 7

that works :-D

Answer 8

yes and it is a valid prove we did it in real analysis