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Example 8.1.7 (b): Pointwise Convergent Function Sequence

Show that fn(x) = xn, x [ 0, 1 ] converges pointwise and identify the limit function.

Let

Then:

  • if x = 1 we have fn(1) = 1 for all n
  • if x < 1 then fn(x) = xn is the power sequence and thus converges to zero

Hence fn(x) f(x) for each fixed x.

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