Given I have a matrix $A\in\mathbb{R}^{n\times n}$ and it is tridiagonal and positive definite such that $b_{ij}$ must be zero if $|i-j|>1$. Furthermore, $0<C_1<a_{ij}<C_2$ for $|i-j|\le 1$.
Could I claim that for $n\ge 2 $, then $C_1<\lambda_{\min}(A)$ and $\lambda_\max(A)<C_2$
The claim does not hold. Consider $$A=\begin{pmatrix} 1+\delta & 1\\ 1& 1+\delta \end{pmatrix},\quad \delta>0 $$ Then $\lambda_\max=2+\delta$ and $\lambda_\min=\delta.$
For the upper bound you need another estimate. For the lower bound you need that diagonal terms are dominating over nondiagonal ones: something of that sort $a_{ii}>|a_{i,i-1}|+|a_{i,i+1}|.$