The sequence {f_n(x)} fulfills all the desired properties:
Each f_n(x) is Riemann integrable on [0, 1].
The sequence converges pointwise to f(x) = 0.
The limit function f(x) is not Riemann integrable.
Here's an example of a convergent sequence of Riemann integrable functions on [0, 1] whose limit function is not Riemann integrable:
Define the sequence:
For each positive integer n, let:
f_n(x) = n * sin(n^2 * pi * x)
Verify its properties:
Riemann integrability of f_n: Each f_n is a continuous function on [0, 1].
Furthermore, it's bounded by n, which implies its integrability using the Riemann integral definition.
Convergence: Consider any point x in [0, 1].
As n approaches infinity, n^2 * pi * x grows unbounded, causing sin(n^2 * pi * x) to oscillate infinitely many times between -1 and 1.
This rapid oscillation makes the sequence converge pointwise to the function:
f(x) = 0
Non-integrability of f: While f(x) = 0 is a simple function, it's not Riemann integrable.
The reason lies in its highly oscillatory behavior for any interval within [0, 1].
Any Riemann sum based on dividing the interval into subintervals will have positive and negative terms that cancel each other out across arbitrarily many subintervals, making the sum converge to different values depending on the chosen Riemann sum method.
This violates the definition of Riemann integrability.
Therefore, the sequence {f_n(x)} fulfills all the desired properties:
Each f_n(x) is Riemann integrable on [0, 1].
The sequence converges pointwise to f(x) = 0.
The limit function f(x) is not Riemann integrable.