Proposition (1). If $f:[a,b]\to\mathbb{R},g:[a,b]\to\mathbb{R}$ are bounded functions and $P:=\{x_0,x_1,x_2,...,x_n\}$ is a partion of $[a,b]$ then $$m_i(f)+m_i(g)\leq m_i(f+g),$$ where for every function $h$, $m_i(h):=\inf\{h(x): x\in[x_{i-1},x_i]\}$.
I want to prove proposition (1) but I have troubles in understanding it. I think the tesis of proposition (1) must be $$m_i(f)+m_i(g)= m_i(f+g),$$ see: $A:=\{f(x):x\in[x_{i-1},x_i\},B:=\{g(x):x\in[x_{i-1},x_i\},C:=\{f(x)+g(x):x\in[x_{i-1},x_i\}$ \begin{align*} &m_i(f)+m_i(g)=m_i(f+g)\\ &\iff \inf(A)+\inf(B)=\inf\{(f+g)(x): x\in[x_{i-1},x_i]\}\\ &\iff \inf(A)+\inf(B)=\inf\{f(x)+g(x): x\in[x_{i-1},x_i]\}, \end{align*} And since $\{f(x)+g(x): x\in[x_{i-1},x_i]\}=A+B$ and $\inf(A)+\inf(B)=\inf(A+B)$, we have that $m_i(f)+m_i(g)=m_i(f+g)$.
Where is my mistake?