Interactive Real Analysis
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Real Analysis
1. Sets and Relations
2. Infinity and Induction
3. Sequences of Numbers
4. Series of Numbers
5. Topology
6. Limits, Continuity, and Differentiation
7. The Integral
8. Sequences of Functions
8.1. Pointwise Convergence
8.2. Uniform Convergence
8.3. Series and Power Series
8.4. Taylor Series
8.5. Approximation Theory
9. Historical Tidbits
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Example 8.1.7 (b): Pointwise Convergent Function Sequence
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Show that
f
_{n}
(x) = x
^{n}
,
x
[ 0, 1 ]
converges pointwise and identify the limit function.
Back
Let
Then:
if
x = 1
we have
f
_{n}
(1) = 1
for all
n
if
x < 1
then
f
_{n}
(x) = x
^{n}
is the
power sequence
and thus converges to zero
Hence
f
_{n}
(x)
f(x)
for each fixed
x
.
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