In the Book "Brownian motion" of Schilling and Partzsch, to prove that $\mathbb E[B_\tau^2]=\mathbb E[\tau]$ whenever $\mathbb E[\tau]<\infty $, they proved that $$(B_{\tau\wedge t})_{t\geq 0},$$ is a $L^2-$Cauchy sequence.
What does this mean since $(B_{\tau\wedge t})_{t\geq 0}$ is even not a sequence ?
I mean, if $n\in\mathbb N$, then $(B_{\tau\wedge n})_n$ is a $L^2-$Cauchy sequence means that $$\lim_{m,n\to \infty }\mathbb E[(B_{\tau\wedge n}-B_{\tau\wedge m})^2]=0,$$
but what does it mean for a continuous process $(B_{\tau\wedge t})_{t\geq 0}$ ? Maybe that $(B_{\tau\wedge t_n})_n$ is a Cauchy sequence for all $(t_n)$ s.t. $t_n\to \infty $ when $n\to \infty $ ?
So, for people who are interested by the proof
I assume you are talking about the proof of Theorem 5.10.
You're right, they are being a little bit sloppy. They are thinking of the limit as $t \to \infty$ as a "sequence", even though the limit should be taken through all the reals.
However, it is easy to fix. Let $t_n$ be any increasing sequence of real numbers which tends to infinity. Then their argument shows that $\{B_{\tau \wedge t_n}\}$ is an $L^2$-Cauchy sequence, and therefore converges to some random variable which we may call $B_\tau$. Moreover, if $t_n'$ is any other increasing sequence which also tends to infinity, their argument also shows that $B_{\tau \wedge t_n} - B_{\tau \wedge t_n'}$ converges to 0 in $L^2$, hence $B_{\tau \wedge t_n'}$ also converges in $L^2$ to the same $B_\tau$.
To finish, we check that indeed $B_{\tau \wedge t} \to B_\tau$ in $L^2$, in the usual sense of limits through the reals: for every $\epsilon > 0$ there exists $M > 0$ such that for all $t > M$ we have $\|B_{\tau \wedge t} - B_\tau\|_{L^2} < \epsilon$. For suppose not. Taking $M = 0$ we can find some $t_1 > 0$ such that $\|B_{\tau \wedge t_1} - B_\tau\|_{L^2} \ge \epsilon$. Taking $M=\max(t_1,1)$ we can find $t_2 > M$ with the same property. Continuing in this way, with $M = \max(t_n, n)$, we find a sequence $t_n$ which is strictly increasing and tends to infinity, such that $\|B_{\tau \wedge t_n} - B_\tau\|_{L^2} \ge \epsilon$ for every $n$, which contradicts the conclusion of the previous paragraph.
This is a good type of argument to know, because it is commonly used to apply results about sequences to limits through the reals.