interchange of limits
An enormous fraction of analysis comes down to one innocent-looking question: can you swap the order of two limiting operations? Limit then integral, or integral then limit? Limit then derivative? Sum then limit? The answer is 'sometimes', and almost every major theorem in this field is really a license to perform one specific swap under stated hypotheses.
The abstract form is a double limit. Given a_{m,n}, the iterated limits lim_m (lim_n a_{m,n}) and lim_n (lim_m a_{m,n}) need not be equal. The function version: lim_{x->a} (lim_n f_n(x)) versus lim_n (lim_{x->a} f_n(x)). When f_n -> f uniformly and each f_n is continuous at a, these agree — that agreement is precisely the uniform limit theorem. Term-by-term integration is the swap of lim_n and integral; term-by-term differentiation swaps lim_n and d/dx.
The honest reality is that limits do NOT commute in general, and the counterexamples are the heart of the subject. The discipline of analysis is largely about cataloguing the SUFFICIENT conditions — uniform convergence, monotonicity (Dini, monotone convergence theorem), domination (dominated convergence theorem) — each of which buys you one legitimate interchange. When in doubt, never swap two limits without a theorem behind you.
Let a_{m,n} = m/(m + n). Then lim_n a_{m,n} = 0 for each m, so lim_m (lim_n a_{m,n}) = 0. But lim_m a_{m,n} = 1 for each n, so lim_n (lim_m a_{m,n}) = 1. The two iterated limits give 0 and 1 — swapping is simply wrong here, with no uniformity to save it.
Iterated limits of m/(m+n): 0 one way, 1 the other.