Suppose $f$ is a measurable function on $\mathbb{R}^d$. Then there exists a sequence of simple functions $\{\varphi_k(x)\}_{k=1}^\infty$ that satisfies $|\varphi_k(x)|\leq|\varphi_{k+1}(x)|$ and $\lim_{k\rightarrow\infty}(x)=f(x)$ for all $x$.
The proof of this goes as follows: decompose $f$ as $f=f^+-f^-$. Since both $f^+$ and $f^-$ are nonnegative, there exist two sequences of increasing nonnegative simple functions $\{f_k^+(x)\}$ and $\{f_k^-(x)\}$ that converge pointwise to $f^+$ and $f^-$, respectively, and are such that $f_k^+(x)\leq f_{k+1}^+(x)$ and $f_k^-(x)\leq f_{k+1}^-(x)$. Then if we let $\varphi_k(x)=f_k^+(x)-f_k^-(x)$, we see that $\varphi_k(x)$ converges to $f(x)$ for all $x$.
The next part of the proof unclear to me. The claim is that the sequence $\{|\varphi_k|\}$ is increasing because $|\varphi_k(x)|=f_k^+(x)+f_k^-(x)$, however why does this equality hold? What properties of the above sequences and functions make this true?