I would like to know the proof of the Weierstrass approximation theorem.
I prefer proof that the story is easy to follow and requires little prior knowledge, but that is somewhat subjective, so I would like to know the various proofs by asking a big-list question (that tag was removed).
Bibliographic information or a summary is sufficient, not the full text of the proof.
Here is a proof for the Weierstrass Approximation Theorem:
Let $f(x)$ be a continuous function on a closed interval $[a, b]$. Then, for any $ε > 0$, there exists a polynomial $p(x)$ such that:
$|f(x) - p(x)|$ $<$ $ε$ for all $x$ in $[a, b]$.
Proof:
Since $f(x)$ is continuous on $[a, b]$, it is uniformly continuous. This means that for any $ε > 0$, there exists a positive number $δ$ such that $|f(x) - f(y)| < ε$ whenever $|x - y| < δ$.
Let's choose a finite set of points ${x_1, x_2, \ldots, x_n}$ in $[a, b]$ such that the intervals $(x_i - δ, x_i + δ)$ for $i = 1, 2,\ldots, n$ cover $[a, b]$. We can do this by dividing [a, b] into subintervals of length $2δ$, and choosing one $x_i$ from each subinterval.
Next, we will use these points to construct a polynomial $p(x)$ that approximates $f(x)$. We use the Lagrange Interpolating Polynomial, which is a polynomial of degree at most $n - 1$ that passes through $n$ points. Specifically, let the Lagrange Interpolating Polynomial through the points $(x_i, f(x_i))$ be given by
$p(x) = \sum_{i=1}^n f(x_i) l_i(x)$
where $l_i(x)$ is the $i-th$ Lagrange basis polynomial, defined as:
$l_i(x) = \prod_{j=1}^n (x - x_j) / (x_i - x_j)$ for $i ≠ j$
$l_i(x) = 1$ for $i = j$
By construction, $p(x)$ passes through the points $(x_i, f(x_i))$, so
$|f(x_i) - p(x_i)| = 0$ for all $i = 1, 2, \ldots, n$
By the triangle inequality,
$|f(x) - p(x)| \leq \sum_{i=1}^n |f(x) - f(x_i)| |l_i(x)|$
Since $f(x)$ is uniformly continuous on $[a, b]$, and each $l_i(x)$ is continuous on $[a, b]$, it follows that $|f(x) - p(x)| < ε$ for all $x$ in $[a, b]$.
This completes the proof. The Weierstrass Approximation Theorem states that any continuous function on a closed interval can be arbitrarily closely approximated by a polynomial.