$\mathbf{Background}$
Define a finite difference grid on the rectangle $[0,X] \times [0,T]$,
$(x_j,t_n) = (x_0 + j\Delta x,n\Delta t) \text { for } 0 \leq j \leq M \text { and } 0 \leq n \leq N , \text { with } \Delta x = \frac{X - x_0}{M}, \text { and } \Delta t = \frac{T}{N}.$
After some manipulations, we obtain
$ A_{j,j-1}U_{j-1}^n + A_{j,j}U_{j}^n + A_{j,j+1}U_{j+1}^{n} = - U_j^{n+1}.$
To be specific, let $U^n$ be the vector $U^n = (u_1^n,\ldots,u_{M-1}^n)$ then the system to be solved is $$AU^n = b^n $$
where $A$ is the $(M-1) \times (M-1)$ tridiagonal matrix with entries
$$A_{j,j-1} = \frac{1}{2} \sigma^2 x_j^2 \frac{1}{\Delta x^2} - rx_j \frac{1}{2 \Delta x}$$ $$A_{j,j} = -\sigma^2 x_j^2 \frac{1}{\Delta x^2} - r - \frac{1}{\Delta t} $$ $$A_{j,j-1} = \frac{1}{2} \sigma^2 x_j^2 \frac{1}{\Delta x^2} + rx_j \frac{1}{2 \Delta x}$$ $\text { for } j = 1,\ldots,M-1.$ Here $r > 0.$
Let $b_j^n = -\frac{1}{\Delta t}u_j^{n+1}.$
Assume that the boundary data $u_0^n$ and $u_M^n$ are non-negative. Also assume that the initial data $u_j^N$ is non-negative. Furthermore, assume that $A_{j,j-1} \geq 0$ for all $j \geq 1.$ Then it follows by inspection that $|A_{jj}| > A_{j,j-1} + A_{j,j+1}$ for all $j \geq 1.$ So, $A$ is diagonally dominant and hence invertible. Hence the values $\{u_j^n\}$ can be computed.
$\mathbf{Question}$
Suppose that at some interior mesh point $(x_k,t_p)$ with $0 < k < M, 0 \leq p < N$ the value $u_k^p$ is bigger than all the other values, i.e. $$u_k ^ p \geq u_j^n \,\,\text { for all } j , n.$$ Since the initial and boundary data are non-negative, we have $u_k^p \geq 0.$ I want to show that this is not possible i.e. there cannot be a value $u_k^p$ which exceeds the maximum value of $u$ at $x_0$, at $x_M = X$ ot $t_n = T.$
In fact, the working has already been shown in the following notes: https://people.math.gatech.edu/~meyer/MA6635/chap4.pdf $\,\,\mathbf {page 16}$ but I'm not being able to understand it.
I understand that $$A_{kk}u_k^p + A_{k,k-1}u_k^p + A_{k,k+1}u_k^{p} = - r u_k^p - \frac{1}{\Delta t}u_k^p < -\frac{1}{\Delta t}u_k^{p+1}.$$ But I don't understand how $$A_{kk}u_k^p + A_{k,k-1}u_{k-1}^p + A_{k,k+1}u_{k+1}^{p} \geq A_{kk}u_k^p + A_{k,k-1}u_k^p + A_{k,k+1}u_k^{p}$$ and why is there inconsistency with the $k$th equation of $AU^p = b^p?$
$\mathbf{Edit}$
I think there is a mistake in the text on page 16. It should be $$A_{kk}u_k^p + A_{k,k-1}u_{k-1}^p + A_{k,k+1}u_{k+1}^{p} \leq A_{kk}u_k^p + A_{k,k-1}u_k^p + A_{k,k+1}u_k^{p}$$ and since , $$A_{kk}u_k^p + A_{k,k-1}u_k^p + A_{k,k+1}u_k^{p} < -\frac{1}{\Delta t}u_k^{p+1}$$ there is inconsistency with the $k$th equation because $$A_{kk}u_k^p + A_{k,k-1}u_{k-1}^p + A_{k,k+1}u_{k+1}^{p} = -\frac{1}{\Delta t}u_k^{p+1}. $$