Is $f_{n}$ is analytic on $(a, b)$ and $f_{n} \rightarrow f$ uniformly on $(a, b)$ then is $f$ analytic on $(a, b)$? 
Is $f_{n}$ is analytic on $(a, b)$ and $f_{n} \rightarrow f$ uniformly
  on $(a, b)$ then is $f$ analytic on $(a, b)$?

Intuitively, I think that the answer is no. I know that the statement holds for integrability and continuity; however, I don't think it's necessary for analyticity. Am I correct?

Stone-Weierstrass theorem indicates that any continuous function $f:[a,b]\to\Bbb R$ is a uniform limit of a sequence of polynomials $p_n$. Polynomials are obviously analytic but $f$ need not be differentiable.
