1a. Prove that the sequence {f𝑛} where n =1 to ∞ does not converge uniformly on [0, ∞): 𝑓𝑛(&#119

[answerhappygod]

1a. Prove that the sequence {f𝑛} where n =1 to ∞ does not
converge uniformly on [0, ∞):
𝑓𝑛(𝑥) =
{ −𝑛𝑥 + 𝑛 , 0 ≤ 𝑥 ≤ 1/𝑛
{0 , 𝑥 > 1/𝑛
b. Note: This is a problem about extending uniform convergence
to a larger domain. Let {𝑓𝑛 }𝑛=1 to ∞ be a sequence of functions on
[0,1] having pointwise limit 𝑓(𝑥) on [0,1]. Further assume that {𝑓𝑛
}𝑛=1 to ∞ converges uniformly on (0,1]. Use the “𝜖 − 𝑁” definition
of uniform convergence to prove that {𝑓𝑛 }𝑛=1 to ∞ converges
uniformly on [0,1].