Mathematics · Real Analysis
Consider the sequence {fn} of continuous functions on [0,1] defined by f1(x) = x/2 and fn+1(x) = fn(x) - 12((fn(x))2 - x) for n = 1, 2, 3, ... Then, which of the following statements is TRUE?
Verified Solution4 OptionsUpdated: October 2026👁️1 View
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- 1fn(x) > √x for all x ∞ (0, 1] and n = 1, 2, 3, ...
- 2√x - fn(x) > 2√x2 + n√x for all x ∞ [0, 1] and n = 1, 2, 3, ...
- 3The sequence {fn} converges uniformly on [0,1] to √x✓
- 4The sequence {fn} converges pointwise but not uniformly on [0,1]
Correct answer
The sequence {fn} converges uniformly on [0,1] to √x
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