While trying to prove that CDFs are right-continuous, I wrote the following proof which seems to actually prove that CDFs are right-continuous if and only if the measure of the given point is zero. I’m certain I’ve made some illicit move, but I can’t spot where.
Let $\mu$ be a finite Borel measure on $\Bbb R$, and define $F(x)=\mu(-\infty,x)$. Let $a\in\Bbb R$. Then $$\lim_{x\to a^+}F(x) = \lim_{x\to a^+} (\mu(-\infty,a)+\mu[a,x)) = F(a)+\lim_{x\to a^+} \mu [a,x)$$ And the limit is the same as $\lim_{n\to\infty}\mu[a,a+1/n)$ which by the continuity of measure is $\mu(\{a\})$. So $F$ is right-continuous if and only if $\mu(\{a\})=0$.
If you define $F(x)=\mu(-\infty, x)$ then it is left-continuous (and is also right-continuous only if $\mu(\{a\})=0$), so your proof is correct. The problem is your definition. Usually the CDF is defined as $F(x)=\mu((\infty,x])$, which is right-continuous.