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The union bound gives $\mathbb{P}({\bigcup }_{n\ge m}{A}_n)\le {\sum }_{n\ge m}\mathbb{P}(A_n)\to 0$. Taking the decreasing intersection proves the first Borel-Cantelli lemma.

If $X_n\to X$ almost surely, then $1_{\{|X_n-X|>\epsilon \}}\to 0$ almost surely; dominated convergence theorem gives convergence of its expectation, hence convergence in probability.

If convergence in probability holds, any subsequence has a further one with $\mathbb{P}(|X_{n(k_r)}-X|>2^{-r})<2^{-r}$. Borel-Cantelli then gives almost-sure convergence. Conversely, failure in probability supplies a subsequence with probabilities bounded below by some positive constant, and no further subsequence can converge almost surely because that would imply convergence in probability.

Choose a nonconstant rational function $t$ on $C$. Functions in $L(D)$ are integral over a finite-dimensional bounded-pole space over $k(t)$; equivalently, evaluation of sufficiently many principal parts embeds $L(D)$ into a finite-dimensional vector space. Hence $L(D)$ is finite dimensional.

If $E=D+\mathrm{div}(h)$, multiplication by $h$ gives the isomorphism $L(E)\to L(D)$. A canonical divisor is the divisor of a nonzero rational differential. Riemann-Roch theorem says $\ell (D)-\ell (K-D)=\mathrm{deg}D+1-g$. Taking $D=0$ gives $1-g=1+g-\mathrm{deg}K$, hence $\mathrm{deg}K=2g-2$.

For a smooth plane curve of degree $d$, the adjunction formula gives $K_C\sim (d-3)H{|}_C$. A cubic has $d=3$, so $K_C\sim 0$.