Given pointwise on a domain , where each is monotone (either increasing or decreasing) on , show that is monotone on .

so what I have is:

assume that f_n(x) is monotone increasing then:

therefore

by definition (the one that I am using) a function is pointwise convergent if and only if and if and only if given an (dependent on both and ) such that

so based on that if I let:

and

therefore so

hence as increase so does

is this correct?