Inequality for certain stochastic process.

Question

Let $b: \mathbb{R} \rightarrow \mathbb{R}$ be a Lipschitz-continuous function and let $X_t$ be a real valued stochastic process satisfying the stochastic differential equation $dX_t= b(X_t) dt+ dB_t$, $X_0=x$. Prove that for any $M> 0$, $t> 0$ and $x \in \mathbb{R}$ we have that $P(X_t \geq M)>0$ but in the case that $b(x)= \alpha$ for some $\alpha <0$ we have that $P(\lim_{t \rightarrow \infty} X_t= - \infty)=1$.