Elementary poof for the commutation of limits and infinite sums Let $\phi_k:\mathbb N \to [0,\infty)$ be a monotonically increasing sequence. ($ \phi_k \le \phi_{k+1}$). Then $\lim_{k \to \infty} \sum_{n=1}^{\infty} \phi_k(n)= \sum_{n=1}^{\infty} (\lim_{k \to \infty} \phi_k)(n)$. This follows from the monotone convergence theorem, when we endow $\mathbb N$ with the counting measure. Is there an elementary proof, in the style fo calculus $1$ or $2$ without using measure theory? We clearly have $\lim_{k \to \infty} \sum_{n=1}^{\infty} \phi_k(n) \le \sum_{n=1}^{\infty} (\lim_{k \to \infty} \phi_k)(n)$. How to prove the reverse inequality, without assuming uniform convergence?

$ \sum\limits_{n=1}^{N} \lim_{k \to \infty} \phi_k(n) = \lim_{k \to \infty} \sum\limits_{n=1}^{N} \phi_k(n) \leq \lim_{k \to \infty} \sum\limits_{n=1}^{\infty} \phi_k(n)$ for each $N$. Let $N \to \infty$.