ca.analysis and odes - erfc lower bound

Durrett, Probability: Theory and Examples, 3rd edition, p. 6 gives

$$(x^{-1} - x^{-3}) e^{-x^2/2} le int_x^infty e^{-y^2/2} : dy $$

The proof Durrett gives is from the observation that

$$ int_x^infty (1-3y^{-4}) e^{-y^2/2} : dy = left( x^{-1} + x^{-3} right) e^{-x^2/2} $$

which I suspect can be found by integration by parts, although I haven't written it out; in any case, differentiate it to check.

After this, some changes of variables give

$$ left( {1 over z} - {1 over 2z^3} right) e^{-z^2}/sqrt{pi} le erfc(z). $$

Finally, $z/(1+2z^2) < 1/z-1/(2z^3)$ for $z > 2^{-1/4}$, giving your bound for $z > 2^{-1/4}$ if $pi$ is replaced with $sqrt{pi}$.

Obviously this is a hack trying to get your proposed bound in the form of the bound I already knew, but hopefully it helps.