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The two preliminary sequence limit laws follow directly from the triangle inequality. For the sum, choose indices so that $|x_n-x|<\epsilon /2$ and $|y_n-y|<\epsilon /2$ beyond their respective cutoffs. Beyond the larger cutoff,Thus

For the reciprocal, convergence to nonzero $x$ gives $|x_n-x|<|x|/2$ eventually. The reverse triangle inequality then gives $|x_n|\ge |x|/2$, and consequentlyChoosing the eventual error smaller than both $|x|/2$ and $\epsilon |x{|}^2/2$ provesThe nonzero limit gives the denominator a positive lower bound; nonzero individual terms alone would not do so. The assumption $x_n\ne 0$ ensures the reciprocal sequence is defined even before this eventual bound applies.

Linearizing the shallow water equations about depth $h$ at rest gives wave speed $c=\sqrt{gh}$. Replacing $h$ by $c$ in the nonlinear equations yields $c_t+u{c}_x+(c/2)u_x=0$ and $u_t+u{u}_x+2c{c}_x=0$. Direct addition and subtraction giveThus the Riemann invariants $J_{\pm }=u\pm 2(c-c_0)$ are constant on the characteristics $C_{\pm }$ with speeds $u\pm c$. In this problem $c_0=\sqrt{g{h}_0}$.