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mathvision
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This is something that comes up when I want to determine whether the sequence of functions {f_n} converge uniformly to f:
Suppose f_n(x) = sqrt(x^2 + 1/n^2), so f(x) = x.
Then, according to Spivak, f(x) - f_n(x) = sqrt(x^2) - sqrt(x^2 + 1/n^2) = 1/(2n^2*sqrt(ε)) for some ε such that x^2 < ε < x^2 + 1/n^2.
Similarly, sqrt(x) - sqrt(x + 1/n) = 1/(2n sqrt(ε)) for some ε such that x < ε < x + 1/n.
Why is this?
I'd really appreciate any help. Thanks!
Suppose f_n(x) = sqrt(x^2 + 1/n^2), so f(x) = x.
Then, according to Spivak, f(x) - f_n(x) = sqrt(x^2) - sqrt(x^2 + 1/n^2) = 1/(2n^2*sqrt(ε)) for some ε such that x^2 < ε < x^2 + 1/n^2.
Similarly, sqrt(x) - sqrt(x + 1/n) = 1/(2n sqrt(ε)) for some ε such that x < ε < x + 1/n.
Why is this?
I'd really appreciate any help. Thanks!