Does the sequence of iterated derivatives of $2^x$ converge uniformly to the zero function on any interval that is bounded above?

Question

This is a follow-up to my previous question, here: Does the sequence of iterated derivatives of $2^x$ converge uniformly to the zero function?. In that question, it was proven that the sequence of iterated derivatives of $2^x$ does not converge uniformly to the zero function over the entire real line. However, does it converge uniformly to the zero function over any interval $(-\infty, a)$? I am pretty sure it does, but I want to see a proof. And, as with my previous question, I used the specific example of the exponential function $2^x$, but I could have used any exponential function $b^x$ with $1 < b < e$.