9. Given the sequence $\left\{a_{n}\right\}$ satisfies the conditions: $a_{1}=1, a_{n+1}=2 a_{n}+1, n \in \mathbf{N}_{+}$.
(1) Find the general term formula for the sequence $\left\{a_{n}\right\}$;
(2) Let $c_{n}=\frac{2^{n}}{a_{n} \cdot a_{n+1}}, T_{n}=\sum_{k=1}^{n} c_{k}$, prove: $T_{n}<1$.
(1) $a_{n+1}+1=2\left(a_{n}+1\right) \Rightarrow a_{n}+1=2^{n} \Rightarrow a_{n}=2^{n}-1\left(n \in \mathbf{N}_{+}\right)$.
(2) $c_{n}=\frac{2^{n}}{\left(2^{n}-1\right)\left(2^{n+1}-1\right)}=\frac{1}{2^{n}-1}-\frac{1}{2^{n+1}-1}$,

So $T_{n}=\sum_{k=1}^{n}\left(\frac{1}{2^{k}-1}-\frac{1}{2^{k+1}-1}\right)=1-\frac{1}{2^{n+1}-1}<1$.