Given a compact Hausdorff (I do not assume metrisability) space $K$ and a sequence $(f_n)_n$ of continuous real-valued functions on $K$ such that the pointwise limit of this sequence exists. Must the limit be Borel-measurable?
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More generally, if $\Sigma$ is a $\sigma$-algebra of subsets of a set $X$, and $(f_n)_n$ is a pointwise convergent sequence of real-valued $\Sigma$-measurable functions on $X$, then $\lim_n f_n$ is $\Sigma$-measurable. (The usual way to prove this is to consider $\liminf$ and $\limsup$.) Since continuous functions are Borel-measurable, the answer to your question is yes.
It is worth mentioning that in the case where $K$ is an interval in $\mathbb R$, what you get is a Baire class 1 function, a very special type of Borel function. A related question asked whether every Lebesgue measurable function is a pointwise limit of continuous functions.