[dagw]

When you have those sorts of problems the best way is to approach them using inequalities.

   3.1415<pi<3.1416 and 1.4142<sqrt(2)<1.4143, => 4.5557<pi + sqrt(2)<4.5559
   => 4.553 < 4.5557 < pi + sqrt(2) => 4.553 < pi + sqrt(2)

[lmm]

Finite decimal representations being of course the most convenient notation for inequalities like this.