Consider the following two proposed simulations of paths of standard, one-dimensional Brownian motion between time $0$ and time $1$.
Normal Increments Roll out a large sequence of, say $M$, independent Gaussian variables $X_1, X_2, X_3, \dots, X_M$ such that $X_n \sim N\left(\mu=0, \sigma^2=\frac{1}{M}\right)$, and plot a scatter plot of the points $(0.001n, \sum_{i = 1}^n X_i)$ with straight lines connecting consecutive points.
Random Walk Trace out the path of a one-dimensional random walk that starts at $0$ and progresses by steps of $\pm \sqrt{\frac{1}{M}}$ every $\frac{1}{M}$th of a second.
Does any of these methods converge to a Brownian motion path? If so, what type of convergence is it?