Real Analysis: Sequences of Functions and Uniform Convergence

What are the uses of non-analytic smooth functions in real analysis?

Given a continuous function on the interval [0, 1], show that Bernstein polynomials converge uniformly to the function as n goes to infinity.

Explain the difference between pointwise and uniform convergence of a sequence of functions.

Given a sequence of continuous functions $f_n$, if their pointwise limit $f$ is not continuous, explain why uniform convergence fails.

Demonstrate why a discontinuous function cannot be uniformly approximated by continuous functions.

Consider the sequence of functions $F_n(x) = x^n$ on the interval $[0,1]$. Determine whether this sequence converges pointwise or uniformly, and identify the limit function.

Prove that the sequence of functions $f_n(x) = \frac{x}{1 + nx^2}$ converges uniformly on the set of real numbers to 0.

Consider a sequence of functions $f_n$ that converges pointwise to a function $f$. Demonstrate why this sequence does not exhibit uniform convergence.

Prove that the series $\sum_{n=1}^{\infty} \frac{1}{1 + n^2} x^4$ is uniformly convergent on the interval $[1, \infty)$.

Check if the sequence of partial sums (SOPS) of a given series of functions is uniformly convergent in the interval [0, 1].

Explain Weierstrass's M-test and its application to determine if a series converges uniformly in a closed interval [a, b].

Using the Weierstrass approximation theorem, find polynomials $p_n$ such that $p_n$ converge to a continuous function $f$ uniformly on the closed interval $[a, b]$.