Let $\Omega = (0,1]$ and $\mathcal{B} = \mathcal{B}((0,1])$. Let $\{b\}$ be a subset of $(0,1]$. The Lebesgue measure $\lambda$ satisfies $\lambda((a,b]) = b - a$ for any $ (a,b] \subseteq (0,1]$
I have been trying to prove that $\lambda(\{b\}) = 0$
So far I have: $\{b\} = \cap_{n=1}^{\infty}[b-1/n,b+1/n]$, since $\lambda((a,b]) = \lambda([a,b])$ we have $\lambda(\cap_{n=1}^{\infty}[b-1/n,b+1/n])= b - b = 0 $
Am I going in the right direction with this proof?