Let us assume that $(4-5\sqrt{2})$ is rational .
Subtract given number from 4, considering 4 is rational number. as we know difference of two rational numbers is rational.
$4-(4-5\sqrt{2})$is rational
$\Rightarrow 5\sqrt{2}$ is rational
which is only possible if 5 is rational and $\sqrt{2}$ is rational.
As we know prouduct of two rational number s rational
But the fact is $\sqrt{2}$ is an irrational.
Which is contradictory to our assumption.
Hence, $4-5\sqrt{2}$ is irrational . hence proved.