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

No. The function f_n(x) is increasing in x, not n. The definition should be that if y>x then f(y)>f(x).
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:

(correct) and

No. This should be