Is this Munkres' proof rigorous? If not, please show me a rigorous proof. 
I am reading "Analysis on Manifolds" by James R. Munkres.  
Is this Munkres' proof of Corollary 10.5 rigorous?
If not, please show me a rigorous proof.
I think Munkres' proof is an intuitive proof.
I think Munkres uses 2-dimensional figure to prove n-dimensional case.
 A: There are deceptively simple theorems that are difficult, e.g., Jordan curve theorem, or just tedious to prove.   Because every step is not justified in detail does not mean the proof is not rigorous.  
Munkres has outlined the steps to prove a seemingly self-evident corollary that the volume of a rectangle is no greater than the sum of volumes of rectangles in a finite cover. That this even is explicitly stated as a corollary speaks to the level of detail that appears in this book -- in contrast to other books on this subject.
A good exercise would be to fill in the details.  Even the first step -- choose a rectangle $Q'$ containing all rectangles $Q_1,Q_2,\ldots, Q_k$ -- requires justification to be completely (and perhaps excessively) thorough.  Supplying the details here should help you do the same for the remainder of the proof.
In this regard, let $Q_j = [a_{j1},b_{j1}] \times \ldots \times [a_{jn},b_{jn}]$ for $1 \leqslant j \leqslant k$, and define 
$$\alpha_p = \min_{1 \leqslant j \leqslant k} a_{jp},\quad \beta_p = \max_{1 \leqslant j \leqslant k} b_{jp}, \quad Q' = [\alpha_1,\beta_1] \times \ldots \times [\alpha_n, \beta_n]$$
If $(x_1,\ldots, x_n) \in Q_j$, then $\alpha_p \leqslant a_{jp} \leqslant x_p \leqslant b_{jp} \leqslant \beta_p$, and consequently $x_p \in [\alpha_p, \beta_p]$ for all $1 \leqslant p \leqslant n$. This, of course, implies that $Q_j \subset Q'$ for all $1 \leqslant j \leqslant k$.
Proceed now to the next step -- use the endpoints of the component intervals of the rectangles $Q, Q_1, \ldots, Q_k$ to define a partition $P$ of $Q'$, etc.
