  We revisit the existence of monotonic quantities along renormalization group
flows using only the Null Energy Condition and the Ryu-Takayanagi formula for
the entanglement entropy of field theories with anti-de Sitter gravity duals.
In particular, we consider flows within the same dimension and holographically
reprove the $c$-, $F$-, and $a$-theorems in dimensions two, three, and four. We
focus on the family of maximally spherical entangling surfaces, define a
quasi-constant of motion corresponding to the breaking of conformal invariance,
and use a properly defined distance between minimal surfaces to construct a
holographic $c$-function that is monotonic along the flow. We then apply our
method to the case of flows across dimensions: There, we reprove the
monotonicity of flows from $\mathrm{AdS}_{D+1}$ to $\mathrm{AdS}_3$ and prove
the novel case of flows from $\mathrm{AdS}_5$ to $\mathrm{AdS}_4$.
