## Proposition 8.1.6: Pointwise Convergence defines Function

There isn't much to prove. We know that since *f _{n}*
converges pointwise, it converges for each fixed

*x*to a limit

*L(x)*. The function defined via

f(x) = L(x)

is indeed a function because ... well, because ... let's see, to be a true
function we would have for each *x* exactly one *f(x) = L(x)*.
But that must be true by some elementary property of numeric sequences.

Which one?