As Brownian motion is a martingale and, in particular, is a semimartingale then its quadratic variation must exist. We just need to compute its value along a sequence of partitions.

If $P=\{0=t_{0}\leq t_{1}\leq\cdots\leq t_{m}=t\}$ is a partition of the interval $[0,t]$, then the quadratic variation on $P$ is

Using the property that the increments $W_{{t_{k}}}-W_{{t_{{k-1}}}}$ are independent normal random variables with mean zero and variance $t_{k}-t_{{k-1}}$, the mean and variance of $[W]^{P}$ are

Here, $|P|=\max_{k}(t_{k}-t_{{k-1}})$ is the mesh of the partition. If $(P_{n})_{{n=1,2,\ldots}}$ is a sequence of partitions of $[0,t]$ with mesh going to zero as $n\rightarrow\infty$ then,

as $n\rightarrow\infty$. This shows that $[W]^{{P_{n}}}\rightarrow t$ in the $L^{2}$ norm and, in particular, converges in probability. So, $[W]_{t}=t$.