assume f_n converges to f pointwise on some open set \(\displaystyle \Omega\) except at one interior point z, and f_n and f are analytic in \(\displaystyle \Omega\). does this imply that f_n converges to f at this point z as well?

it seems it would be just crazy if it didn't converge, but i can't prove it.

i also need the same statement for meromorphic f_n and f, but i suspect the answer would be the same.