Problem 11.2.7. Prove Theorem 11.2.6. Theorem 11.2.6. Suppose for every n E N fn is differentiable, fis unif →g on an interval, I. Then f is differentiable ptwise continuous, fnf, and f and f' = g on I. Hint. in-context

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Chapter2: Second-order Linear Odes
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Theorem 11.2.6. Suppose for every n E N fn is differentiable, f is continuous,
ptwise
unif
f, and fr g on an interval, I. Then f is differentiable and f' = g on I.
fn
Problem 11.2.7. Prove Theorem 11.2.6.
Theorem 11.2.6. Suppose for every n E N fn is differentiable, f is
unif
→g on an interval, I. Then f is differentiable
ptwise
continuous, fn →f, and f
and f' = g on I.
in-context
▼ Hint.
Let a be an arbitrary fixed point in I and let x = I. By the Fundamental
Theorem of Calculus, we have
x
[ª_ f(t)dt = fn(x) — fn(a).
t=a
Take the limit of both sides and differentiate with respect to x.
Transcribed Image Text:Theorem 11.2.6. Suppose for every n E N fn is differentiable, f is continuous, ptwise unif f, and fr g on an interval, I. Then f is differentiable and f' = g on I. fn Problem 11.2.7. Prove Theorem 11.2.6. Theorem 11.2.6. Suppose for every n E N fn is differentiable, f is unif →g on an interval, I. Then f is differentiable ptwise continuous, fn →f, and f and f' = g on I. in-context ▼ Hint. Let a be an arbitrary fixed point in I and let x = I. By the Fundamental Theorem of Calculus, we have x [ª_ f(t)dt = fn(x) — fn(a). t=a Take the limit of both sides and differentiate with respect to x.
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