= Solution
{c}
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 variance
$$
\left(\frac{\sigma_1^2}{\sigma^2}+\frac{\sigma_2^2}{\sigma^2}\right)h=h.
$$
Increments on disjoint intervals are independent, since they depend on disjoint increments of the two independent <Brownian motions>. These are the defining properties, so \b[$W_3$ is a standard Brownian motion].
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