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Example 8.1.7 (a): Pointwise Convergent Function Sequence

Let fn(x) = max(n - n2 |x - 1/n|, 0), x [ 0, 1 ]. Show that this sequence converges pointwise to the function f(x) = 0 for x [ 0, 1 ].

Recall that for each n the function fn(x) = max(n - n2 |x - 1/n|, 0) is piecewise linear:
Now take any > 0. Fix x [ 0, 1 ].
  • if x = 0 the sequence fn(0) is identically zero, hence converges to zero trivially
  • if x > 0 find an integer N such that 2/N < x. Then
    | fn(x) - 0 | = 0 < for n > N
    Hence the sequence converges to zero again.
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