and (3.22), we have The claim follows from taking the limit as δ ↓ 0. Claim 3.3.7.3. For each δ > 0 , there exist integral simplicial chains Y$_{δ}$ and R$_{δ}$ on K$_{δ}$ such that P$_{δ}$ = Y$_{δ}$ + ∂R$_{δ}$ is an optimal simplicial flat norm decomposition (i.e., F$_{K}$$_{δ}$ ( P$_{δ}$ ) = M( Y$_{δ}$ ) + M( R$_{δ}$ ) ). Proof of claim. This follows from Theorem 3.2.3. Claim 3.3.7.4. There exists c > 0 such that for all δ ≤ 1 , the currents Y$_{δ}$ , ∂Y$_{δ}$ , R$_{δ}$ , and ∂R$_{δ}$ all have mass at most c . Proof of claim. Using the fact that P$_{δ}$ = Y$_{δ}$ + ∂R$_{δ}$ is an optimal simplicial flat norm decomposition and facts from Equation (3.17), we observe that 76 M( Y$_{δ}$ ) ≤ M( P$_{δ}$ ) < M( T ) + δ ≤ M( T ) + 1 , M( R$_{δ}$ ) ≤ M( P$_{δ}$ ) < M( T ) + δ ≤ M( T ) + 1 , M( ∂Y$_{δ}$ ) = M( ∂ ( P$_{δ}$ − ∂R$_{δ}$ )) = M( ∂P$_{δ}$ ) < M( ∂T ) + δ ≤ M( ∂T ) + 1 , M( ∂R$_{δ}$ ) = M( P$_{δ}$ − Y$_{δ}$ ) ≤ M( P$_{δ}$ ) + M( Y$_{δ}$ ) < 2 M( T ) + 2 . F$_{K}$$_{δ}$ ( P$_{δ}$ ) ≤ M( X$_{δ}$ + U$_{0}$$_{,δ}$ + U$_{1}$$_{,δ}$ ) + M( S$_{δ}$ + W$_{0}$$_{,δ}$ + W$_{1}$$_{,δ}$ + W$_{2}$$_{,δ}$ ) ≤ M( X$_{δ}$ ) + M( U$_{0}$$_{,δ}$ ) + M( U$_{1}$$_{,δ}$ ) + M( S$_{δ}$ ) + M( W$_{0}$$_{,δ}$ ) + M( W$_{1}$$_{,δ}$ ) + M( W$_{2}$$_{,δ}$ ) ≤ M( X ) + M( S ) + 2 δ + 2(11 L ) p − $^{d}$(1 + 11 L ) δ + 3(11 L ) p − d − $^{1}$(1 + 11 L ) δ = F ( T ) + 2 δ + 2(11 L ) p − $^{d}$(1 + 11 L ) δ + 3(11 L ) p − d − $^{1}$(1 + 11 L ) δ.