8.1. Pointwise Convergence

Example 8.1.4: A Function Sequence

Define { fn(x) }, where
fn(x) = max(n - n2 |x - 1/n|, 0)
and x [ 0, 1 ]. Describe the elements of this function sequence.


A few family members for n=1 to n=10
For each n the function fn(x) = max(n - n2 |x - 1/n|, 0) is piecewise linear:
If 0 x 1/n:
fn(x) = n - n2(1/n - x) = n2 x i.e. linear with slope n2. The maximum is fn(1/n) = n.
If 1/n x 2/n:
fn(x) = n - n2(x - 1/n) = 2n - n2 x i.e. linear with slope -n2, from fn(1/n) = n to fn(2/n) = 0.
If 2/n x 0:
fn(x) = 0 because of the 'max' function.
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