Description
For each n in N, let gn(x) be defined for x => 0 by the formula
gn(x) = nx , 0 <= x <= ( 1/n )
= (1/ nx), 1/n < x
show that lim( gn (x) = 0 ) for all x > 0 and is not uniform on the domain x=> 0, but that it is uniform on a set x => c where c > 0.
Explanation & Answer
The solutions are here.
𝑛𝑥, 0 ≤ 𝑥 ≤
𝑔𝑛 (𝑥) = {
1
,
𝑛𝑥
1
,
𝑛
1
< 𝑥.
𝑛
0. ∀𝑥 ≥ 0 𝑔𝑛 (𝑥) → 0, 𝑛 → ∞.
𝑥≥0
1. It is not true that 𝑔𝑛 (𝑥) ⇉ 0.
𝑛→∞
𝑥≥𝑐
2. It is true that ∀𝑐 > 0 𝑔𝑛 (𝑥) ⇉ 0.
𝑛→∞
0. For 𝑥 = 0 𝑔�...
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