= Solution
We prove the <Birkhoff–Grothendieck theorem> by induction on the rank of the <vector bundle> $E$. Rank zero is immediate; rank one is the classification $\operatorname{Pic}(\mathbb P^1_k)\cong\mathbb Z$ established above.
First there is a largest integer $a$ with $H^0(E(-a))\ne0$. To see existence and boundedness directly, trivialize $E$ on the two standard affine charts: the associated finite projective modules over the <principal ideal domains> $k[t]$ and $k[t^{-1}]$ are free. Let $A(t)\in GL_r(k[t,t^{-1}])$ be its transition matrix, so a section of $E(-a)$ is a pair satisfying
$$
p(t)=t^{-a}A(t)q(t^{-1}),\qquad p\in k[t]^r,\quad q\in k[t^{-1}]^r.
$$
If $a$ exceeds the largest exponent in any entry of $A$, the right side has only strictly negative powers; the left side has nonnegative powers. Hence both vanish. For $a$ sufficiently negative, take a nonzero constant vector $p$ and put $q=t^aA(t)^{-1}p$; all its exponents are nonpositive, so it supplies a nonzero section. Thus a largest $a$ exists.
Such a section gives a nonzero map $\mathcal O(a)\to E$. It is injective, since it is nonzero generically and the source is torsion-free. Saturate its image. The resulting <line subbundle> is $\mathcal O(a+d)$ for the degree $d\geq0$ of the <effective divisor> of zeros of the original map. If $d>0$, its inclusion in $E$ would give a nonzero section of $E(-a-d)$, contradicting maximality. Therefore $d=0$ and
$$
0\longrightarrow\mathcal O(a)\longrightarrow E\longrightarrow Q\longrightarrow0
$$
has a <locally free> quotient. By induction, $Q\cong\bigoplus_j\mathcal O(b_j)$.
Twist this exact sequence by $\mathcal O(-a-1)$. The outer groups $H^0(\mathcal O(-1))$ and $H^1(\mathcal O(-1))$ are zero by the <Čech cohomology of twists on the projective line>. The <long exact sequence in sheaf cohomology> therefore gives
$$
H^0(E(-a-1))\cong H^0(Q(-a-1)).
$$
The first is zero by the choice of $a$, so each $H^0(\mathcal O(b_j-a-1))$ vanishes. Thus $b_j\leq a$ for every $j$. The extension class lies in
$$
\operatorname{Ext}^1(Q,\mathcal O(a))\cong\bigoplus_jH^1(\mathcal O(a-b_j))=0,
$$
since $a-b_j\geq0$. The identification of this <Ext functor> with cohomology can be seen without additional duality: local splittings of a vector-bundle extension differ by a <Čech cocycle> in $\mathcal H om(Q,\mathcal O(a))$, and a <coboundary> changes them to compatible splittings. Vanishing therefore gives a global splitting. Consequently
$$
\boxed{E\cong\mathcal O(a)\oplus\bigoplus_j\mathcal O(b_j),}
$$
completing the induction over any <field> $k$.
The degrees can be arranged in nonincreasing order and their multiset is unique. Indeed, for any such decomposition,
$$
h^0(E(m))-h^0(E(m-1))=\#\{i:a_i\geq-m\},
$$
so the dimensions of the twisted <global sections> recover the number of summands of every degree.
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