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For a line $L$ on a degree-$d$ smooth surface in ${\mathbb{P}}^3$, the adjunction formula gives $L^2=2-d$. If $d\ge 3$, all lines are negative curves, hence there are only countably many by countability of negative curves. Their projective Fano scheme of lines cannot have a positive-dimensional complex component, since such a component has uncountably many points. A zero-dimensional projective scheme of finite type has finitely many points, proving the assertion.

Let $X\subset {\mathbb{P}}^3$ have degree $d\ge 3$ and hyperplane class $H$. The adjunction formula gives $K_X=(d-4)H$. For a line $L\subset X$, the same formula on the smooth rational curve $L$ yields $(K_X+L)\cdot L=-2$, henceThus all lines are negative curves, and part (b) makes their set countable. The Fano scheme of lines is a closed projective scheme of $\mathrm{Gr}(2,4)$. A positive-dimensional complex algebraic component would have uncountably many points, so this scheme is zero-dimensional. A projective scheme of finite type has only finitely many zero-dimensional points. This proves the finiteness of lines on a smooth surface of degree at least three.