- #1
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Homework Statement
Determine the convergence, both pointwise and uniform on [0,1] for the following sequences :
(i) ##s_n(x) = n^2x^2(1 - cos(\frac{1}{nx})), x≠0; s_n(0) = 0##
(ii) ##s_n(x) = \frac{nx}{x+n}##
(iii) ##s_n(x) = nsin(\frac{x}{n})##
Homework Equations
##s_n(x) → s(x)## as ##n→∞##
The Attempt at a Solution
(i) So for this one, I can split it into a piecewise function since ##s_n(x)## has been defined at the origin.
Taking the limit as n → ∞, I can observe two limiting functions occurring. s(x) = 0 and s(x) = 1/2.
Therefore ##s_n(x)## converges pointwise, but not uniformly on [0,1].
(ii) As n → ∞ ##s_n(x) → x = s(x)## for all x in [0,1]. Hence ##s_n(x)## converges pointwise AND uniformly on [0,1].
(iii) Exact same limit as (ii) so the same answer will follow.
Do these look okay? It seems way too straightforward so it has me a bit worried.
If anyone could confirm it would be great :). Thanks.