When $N=\mathbb{R}$, this is a classical result by Friedrichs and Gårding :

Theorem. (1) If a sequence of real-valued harmonic functions converges uniformly on compact subsets of $M$, then the limit of the sequence is an harmonic function.

(2) Furthermore, the derivatives of the members of the sequence converge uniformly on compact sets to the derivatives of the limit of the sequence.

See p. 220 of

Greene, R. E.; Wu, H., Embedding of open Riemannian manifolds by harmonic functions, Ann. Inst. Fourier 25, No. 1, 215-235 (1975). ZBL0307.31003.

When $N=\mathbb{R}$, this is a classical result by Friedrichs and Gårding :

Theorem. (1) If a sequence of real-valued harmonic functions converges uniformly on compact subsets of $M$, then the limit of the sequence is a harmonic function.

(2) Furthermore, the derivatives of the members of the sequence converge uniformly on compact sets to the derivatives of the limit of the sequence.

See p. 220 of

Greene, R. E.; Wu, H., Embedding of open Riemannian manifolds by harmonic functions, Ann. Inst. Fourier 25, No. 1, 215-235 (1975). ZBL0307.31003.