= Solution
The <Weierstrass approximation theorem> asserts that every real <continuous function> on a <compact> <interval> is a <uniform limit> of <polynomials>. An affine change of variable reduces the interval to $[0,1]$. There are two complementary routes: construct positive approximation operators, or construct an algebra dense enough to approximate absolute values and hence pointwise maxima and minima.
For the first route, the <Korovkin theorem> reduces convergence of <positive linear operators on continuous functions> $L_n$ to the three tests $1,t,t^2$. Here is its proof. Given $f\in C[0,1]$ and $\varepsilon>0$, <uniform continuity> supplies a $\delta>0$ such that $|f(t)-f(x)|<\varepsilon$ for $|t-x|<\delta$. For all $t,x$ one then has
$$
|f(t)-f(x)|\le\varepsilon+M(t-x)^2,\qquad M=2\|f\|_\infty/\delta^2.
$$
Apply positivity to the two real inequalities obtained from this bound. At $x$ this gives
$$
|L_nf(x)-f(x)|\le |f(x)|\,|L_n1(x)-1|+\varepsilon L_n1(x)+M\bigl[L_nt^2(x)-2xL_nt(x)+x^2L_n1(x)\bigr].
$$
The expression in brackets is nonnegative and tends uniformly to zero by the three test assumptions. Also $L_n1\to1$ uniformly. Taking the upper limit of the <supremum norm> and then letting $\varepsilon\downarrow0$ proves convergence for every $f$. This is the <quadratic barrier estimate for positive approximation operators>.
The <Bernstein polynomial> supplies the constructive corollary
$$
B_nf(x)=\sum_{j=0}^nf(j/n)\binom nj x^j(1-x)^{n-j}.
$$
The <binomial theorem> makes its nonnegative weights sum to one. Differentiating the <binomial theorem>, or using the first two <binomial distribution> moments, gives
$$
B_n1=1,\qquad B_nt=x,\qquad B_nt^2=x^2+\frac{x(1-x)}n.
$$
The <Korovkin theorem> proves $B_nf\to f$ uniformly, and each $B_nf$ is a <polynomial>, proving the <Weierstrass approximation theorem>. The same criterion gives the <Schoenberg spline operator> convergence in Solution 3. For periodic <functions> its counterpart uses $1,\cos t,\sin t$: the nonnegative barrier $1-\cos(t-x)$ is bounded away from zero outside any fixed small arc. This <periodic Korovkin test set> gives convergence of the normalized <Fejér sums> and density of <trigonometric polynomials>.
The <Lebesgue convergence criterion for approximation operators> addresses stability. Suppose bounded <linear operators> $L_n:C[a,b]\to C[a,b]$ converge to the identity on every <polynomial>. Then they converge on every <continuous function> if and only if $\sup_n\|L_n\|<\infty$. Sufficiency is the <extension of approximation operators from a dense subspace>: for a <polynomial> $p$ with $\|f-p\|<\eta$ and a common bound $M$,
$$
\|L_nf-f\|\le(M+1)\eta+\|L_np-p\|.
$$
First let $n\to\infty$, then $\eta\downarrow0$. Necessity follows from the <Uniform boundedness principle>, since a convergent $L_nf$ is bounded for every $f$. Applied to approximation <projectors>, the <Lebesgue inequality for approximation projectors> is
$$
\|f-P_nf\|_\infty\le(1+\|P_n\|)E_{V_n}(f),\qquad E_{V_n}(f)=\inf_{v\in V_n}\|f-v\|_\infty.
$$
Indeed $P_nv=v$ makes $f-P_nf=(f-v)-P_n(f-v)$; take the infimum. Thus density of the ranges and a uniform bound on the <Lebesgue constants of interpolation> imply convergence. Density alone does not control a sequence of interpolation <projectors>.
There is also a kernel version of this stability criterion, useful for <trigonometric polynomial> approximation. If <integrable> periodic kernels $K_n$ have uniformly bounded $L^1$ <norms>, their integrals tend to one, and $\int_{|t|\ge\delta}|K_n(t)|\,dt\to0$ for every $\delta>0$, then convolution with $K_n$ converges uniformly to every <continuous> periodic <function>. Subtract $f(x)\int K_n$ and split the error into a small arc, bounded by $\omega(f,\delta)\sup_n\|K_n\|_1$, and its complement, bounded by $2\|f\|_\infty\int_{|t|\ge\delta}|K_n|$. The remaining mass error tends to zero. This <approximate identity> argument shows directly why nonnegative unit-mass <Fejér kernels> work even though ordinary <Fourier partial sums> need not converge uniformly for every continuous target.
Lebesgue's constructive approach to the <Weierstrass approximation theorem> starts instead with the <absolute value>. On $[-1,1]$ define the <polynomial recurrence for the absolute value>
$$
p_0(x)=0,\qquad p_{r+1}(x)=p_r(x)+\frac{x^2-p_r(x)^2}{2}.
$$
Induction gives $0\le p_r(x)\le|x|$, monotonicity, and
$$
|x|-p_{r+1}(x)=(|x|-p_r(x))\left(1-\frac{|x|+p_r(x)}2\right).
$$
The pointwise limit satisfies $p(x)^2=x^2$ and is nonnegative, hence equals $|x|$. <Dini's theorem> makes the convergence uniform. More explicitly, the continuous decreasing errors have, for any positive tolerance, open sublevel sets covering the <compact> <interval>; a finite subcover supplies one index valid everywhere. Every <continuous> <piecewise linear function> has the form
$$
L(x)=a+bx+\sum_{j=1}^mc_j(x-t_j)_+,\qquad (x-t)_+=\frac{x-t+|x-t|}{2}.
$$
The $c_j$ are its changes in slope. Scaling and translating the absolute-value approximation shows that $L$ is uniformly approximable by <polynomials>. Solution 4, or polygonal interpolation and <uniform continuity>, makes these <piecewise linear functions> dense in $C[a,b]$. Approximating $f$ first by $L$ and then by a <polynomial> is another complete proof of the <Weierstrass approximation theorem>.
The <Stone-Weierstrass theorem> generalizes from intervals to a <compact Hausdorff space> $K$: a real subalgebra $A\subseteq C(K,\mathbb R)$ containing the constants and separating points is uniformly dense. Its proof uses the same absolute-value mechanism. The <uniform closure of a function algebra> is an algebra. If $a$ belongs to this closure, approximation of $|t|$ by <polynomials> on a bounded interval containing its range proves that $|a|$ also belongs to the closure. Therefore the closure contains
$$
\max(a,b)=\frac{a+b+|a-b|}{2},\qquad\min(a,b)=\frac{a+b-|a-b|}{2}.
$$
Point separation and constants allow exact interpolation of any two real values: choose $h\in A$ with $h(s)\ne h(t)$ and solve for $\alpha,\beta$ in $\alpha h+\beta$. The <lattice approximation from two-point approximation> proved in Solution 4 now makes the closure all of $C(K,\mathbb R)$.
The complex version requires a unital, point-separating subalgebra closed under <complex conjugation>. Its real-valued members separate points using real or imaginary parts, so the real result approximates both parts of every complex target. Conjugation closure is essential: <polynomials> in $z$ alone on the closed unit disk cannot approximate $\overline z$ uniformly, because $\int_{|z|=1}p(z)\,dz=0$ whereas $\int_{|z|=1}\overline z\,dz=2\pi i$.
Among the corollaries are \b[density of multivariate polynomials on compact subsets of real Euclidean space], since coordinates separate points, and \b[density of trigonometric polynomials on the circle], since sine and cosine separate points there. On a <compact> <interval>, <polynomials> with rational <coefficients> are a countable dense family, giving separability of $C[a,b]$. The real and complex approximation results above concern uniform approximation of continuous targets; neither asserts convergence of arbitrary interpolation schemes or arbitrary <Fourier series>.
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