Brownian motion with a stopping time

Question

Let $x \geq 0,c<0,$ and a Brownian motion $(W_u)_{u}.$ Let $T:=\inf\{u \geq 0, B_u +cu\geq x\}.$

It follows that $Y:=\sup_{u \geq 0}(B_u+cu) \in ]0,\infty[.$

We want to verify that $\{Y \geq x\} \subset (T<\infty).$

Supposing that $Y(w) > x$ then the result follows.

But what if $Y(w)=x,$ I can't see how to deduce it ? Do we need to use the continuity of the BM?

Answer 1

Suppose $T=\infty$. Then $B_u+cu <x$ for all $u$ and $B_{u_n}+cu_n \to x$ for some sequenece $u_n \to \infty$ (because $Y=x$). But the $\frac {B_{u_n}} {u_n}+c \to 0$. This is a contradiction becasue $\frac {B_{u_n}} {u_n} \to 0$ (a.s.)and $c <0$.