What does "the sequence of functions has limit in R" mean?

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SUMMARY

The discussion clarifies the concept of a sequence of functions from a set X to the real numbers R, specifically focusing on the condition that the sequence has a limit in R. The set T is defined as T={x in X: f1(x),... has a limit in R}, indicating that for each x in X, the sequence of function values converges to a single real number. This means that while the limit is a numerical value for each specific x, it collectively defines a function across the entire set X.

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cbarker1
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TL;DR
I am trying understand a condition in a set notion.
Suppose f1,f2... is a sequence of functions from a set X to R. This is the set T={x in X: f1(x),... has a limit in R}. I am confused about what is the meaning of the condition in the set. Is the limit a function or a number value? Why?
 
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I would have to say both. The condition "f1(x),... has a limit in R" is applied for one value of ##x \in R## at a time, so it gives a single value. But it is applied for every ##x \in X##, so that gives a function on ##X##.
 
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