I am reading Gregory F. Lawler's Random Walk and the Heat Equation. In page 39-40 the author considers a set the following problem: Let $N$ be a positive integer, $N\geq 2$. Let $A_N=\{(x_1,x_2): x_i=1,\ldots, N-1\}\subset \mathbb{Z}^2$. For each $j\in \{1, \ldots , N-1\}$, it is easy to see that there is a unique positive number $\beta_j$ such that $$ \cosh(\frac{\beta_j}{N})+\cos(\frac{j\pi}{N})=2. $$ Now for any $y\in \{1, \ldots , N-1\}$ and $(x_1,x_2)\in A_N$ we can define a function $$ H_{(N,y)}(x_1,x_2)=\frac{2}{N-1}\sum_{j=1}^{N-1}\frac{\sin(\frac{j\pi y}{N})\sinh(\frac{\beta_j x_1}{N})\sin(\frac{j\pi x_2}{N})}{\sinh(\beta_j)}. $$ We can consider $H_{(N,y)}(x_1,x_2)$ as a "discrete harmonic function" on $A_N$.
Now consider a subset $\hat{A}_N$ of $A_N$ defined as $$ \hat{A}_N=\{(x_1,x_2)\in A_N: \frac{N}{4}\leq x_1,x_2\leq \frac{3N}{4}\}. $$
Then the author leaves as an exercise to prove that there exists two constants $c$, $c_1<\infty$, independent of $N$, such that for any $y=1,\ldots, N-1$ and any $x$, $\tilde{x}\in \hat{A}_N$ we have $$ \frac{\sin(\frac{\pi y}{N})}{cN}\leq H_{N,y}(x)\leq \frac{c \sin(\frac{\pi y}{N})}{N} $$ and $$ |H_{N,y}(x)-H_{N,y}(\tilde{x})|\leq c_1 \frac{|x-\tilde{x}|}{N}H_{N,y}(x). $$
How to prove these two inequalities?