= Solution
<Minkowski inequality> states that on a <measure space> $(X,\Sigma,\mu)$, for $1\leq p\leq\infty$ and measurable real or complex $f,g$ of finite <Lp norm>,
$$
\boxed{\|f+g\|_p\leq\|f\|_p+\|g\|_p.}
$$
For $p=1$, integrate the scalar <triangle inequality> $|f+g|\leq|f|+|g|$. For $p=\infty$, the <almost everywhere> bounds $|f|\leq\|f\|_\infty$ and $|g|\leq\|g\|_\infty$ give the same result by taking the <essential supremum>.
For $1<p<\infty$, first ensure that the left side is finite. The <convex function> $s\mapsto s^p$ on $[0,\infty)$ gives
$$
|f+g|^p\leq(|f|+|g|)^p\leq2^{p-1}(|f|^p+|g|^p),
$$
so $f+g$ has finite <Lp norm>. Set $q=p/(p-1)$ and $H=\|f+g\|_p$. Since $(p-1)q=p$, <Holder inequality> gives
$$
H^p\leq\int_X(|f|+|g|)|f+g|^{p-1}\,d\mu
\leq(\|f\|_p+\|g\|_p)\left(\int_X|f+g|^p\,d\mu\right)^{1/q}
=(\|f\|_p+\|g\|_p)H^{p-1}.
$$
If $H=0$ the result is immediate; otherwise divide by $H^{p-1}$. This proves the <triangle inequality> for every claimed exponent.
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