Interactive Real Analysis

• 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
• Java Tools

8.1. Pointwise Convergence

Example 8.1.7 (a): Pointwise Convergent Function Sequence

Let f_n(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 ].

Back

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.