Consider $n\times n$ real matrices $A$ and $B$.
If $|A|_m$ denotes the modulus matrix of $A=[a_{i,j}]_{n\times n}$, and is defined as $|A|_m := [|a_{i,j}|]_{n\times n}$, prove that $|A+B|_m\leq|A|_m+|B|_m$.
(We mean by $N\leq M$ that matrix $(N-M) $ is a negative semi-definite matix)
I have seen this inequality in a paper, where no proof is provided.
As you use positive semi-definite partial ordering, presumably $A$ and $B$ are symmetric matrices. Without imposing further conditions, the inequality you want to prove is wrong. Counterexample: $$ A=\pmatrix{2&1\\ 1&2},\ B=\pmatrix{2&-1\\ -1&2}. $$ We have $$ |A|_m+|B|_m-|A+B|_m=\pmatrix{0&1+1-0\\ 1+1-0&0}=\pmatrix{0&2\\ 2&0}, $$ which is indefinite.
The inequality is true, though, if the matrices are compared entrywise. As indicated by the other answer, this is just a simple consequence of the triangle inequality.