Noise predictors have negative OOS $R^2$

Excess returns are $r_{t+1} = \mu + \varepsilon_{t+1}$ with $\varepsilon$ i.i.d. $(0,\sigma_\varepsilon^2)$, and $x_t$ is an i.i.d. $(0,\sigma_x^2)$ "predictor" independent of all $\varepsilon$. Assume $\mu$ is known, so the benchmark forecast is $\mu$. (a) For the fixed forecast $\hat r_{t+1} = \mu + c\,x_t$, show that the population out-of-sample $R^2$ is $-c^2\sigma_x^2/\sigma_\varepsilon^2$. (b) Now $c$ is replaced by the OLS slope $\hat\beta$ from an independent training sample of size $T$ (regressing $r - \mu$ on $x$ without intercept). Show that, to first order, $\mathbb E[R^2_{OS}] \approx -1/T$. What does this imply for a researcher who tries $k$ noise predictors in one multiple regression?