Tangency portfolio and CML - Why does it have the highest sharpe ratio?

In the book that I am studying, the tangent portfolio was defined as the regular efficient portfolio in the case with $n$ risky assets and 1 riskfree asset with the extra requirement that the portfolio invests fully in the risky assets. So the tangent portfolio can be derived using the solutions to the mean/variance analysis problem:

$$w = \frac{\mu_P}{\mu^T \Sigma^{-1} \mu}\Sigma^{-1} \mu$$ $$\sigma_P^2 = \frac{\mu_P^2}{\mu^T \Sigma^{-1} \mu}$$ where one can apply the restrictions on $w$ to obtain weights, mean excess return, and variance of the portfolio.

Yet I know that in other books, this portfolio is actually defined as the one with the highest sharpe ratio. I don't see the connection. How is this proven, if we used the derivation described above? I can calculate the sharpe ratio (it turns out to be the square root of the denominator in the second equation above), but how do I know it's bigger than the ones corresponding to all other investments in risky assets?