So c = max { 2 M( T ) + 2 , M( ∂T ) + 1 } works. Claim 3.3.7.5. There is an optimal flat norm decomposition of T with integral currents. Proof of claim. The compactness theorem [19,32] states that given any closed ball K in R n and nonnegative constant c , the set is compact with respect to the flat norm. In light of Claim 3.3.7.4, this means there is a compact set of integral currents containing Y$_{δ}$ for all δ ≤ 1 (and similarly for R$_{δ}$ ). Let δ$_{n}$ = 1 n and consider the sequences { Y$_{δ}$$_{n}$ } and { R$_{δ}$$_{n}$ } . By compactness, there exists a subsequence { δ ∗ n } of { δ$_{n}$ } and integral currents Y ∗ and R ∗ such that Y$_{δ}$ ∗ n → Y ∗ and R$_{δ}$ ∗ n → R ∗ in the flat norm. By Lemma 3.2.2, we have Y$_{δ}$ ∗ n + ∂R$_{δ}$ ∗ n → Y ∗ + ∂R $^{∗}$. Applying Claim 3.3.7.3 and Claim 3.3.7.2, we get M( Y$_{δ}$ ∗ $_{n}$)+M( R$_{δ}$ ∗ $_{n}$) = F$_{K}$$_{δ}$ ( P$_{δ}$ ∗ $_{n}$) → F ( T ). Since Y$_{δ}$ + ∂R$_{δ}$ = P$_{δ}$ → T , we also have Y$_{δ}$ ∗ n + ∂R$_{δ}$ ∗ n → T . That is, T = Y ∗ + ∂R $^{∗}$. As mass is lower semicontinuous with respect to convergence in the flat norm and by Claim 3.3.7.2, we have that Thus M( Y $^{∗}$) + M( R $^{∗}$) = F ( T ) and Y ∗ + ∂R ∗ is an optimal flat norm decomposition of T . 77 M( Y $^{∗}$) + M( R $^{∗}$) ≤ lim n $_{→∞}$M( Y$_{δ}$ ∗ $_{n}$) + M( R$_{δ}$ ∗ $_{n}$) = lim n →∞ F$_{K}$$_{δ}$$_{∗}$ $_{n}$( P$_{δ}$ ) = F ( T ) . { I is an integral p -current in R n | M( I ) ≤ c, M( ∂I ) ≤ c, spt I ⊆ K }