# What is meant by the word ‘control’ in the context of analysis?

Wikipedia’s article on Dini’s Theorem states:

This is one of the few situations in mathematics where pointwise convergence implies uniform convergence; the key is the greater control implied by the monotonicity.

I’ve heard the word ‘control’ used like this a few times before, but I can’t get a good handle on what it means.

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In your case, the assumption of monotone convergence transforms pointwise convergence into uniform convergence. In general, the pointwise limit of a sequence of continuous $\mathbb{R}$-valued functions on a topological space is a pretty wild animal. When the topological space is a metric space, we call such a function a ‘Baire-$1$ function’. Baire-$1$ functions can be hard to visualize. However, with the assumption of monotone convergence, the pointwise limit of a sequence of continuous $\mathbb{R}$-valued functions on a compact space will always be a nice continuous $\mathbb{R}$-valued function. One can thus say that the assumption of monotone convergence controls the behavior of pointwise limits.