 Solution

Put $\sigma =\sqrt{{\sigma }_1^2+{\sigma }_2^2}>0$. The linear combination $W_3$ starts at zero and has continuous paths. An increment over an interval of length $h$ is a linear combination of independent centred normal random variables, so is centred normal with varianceIncrements on disjoint intervals are independent, since they depend on disjoint increments of the two independent Brownian motions. These are the defining properties, so $W_3$ is a standard Brownian motion.