# Proof about integers and rationals

let x,y,z,w be rational numbers

a) Let epsilon>0. If x is epsilon-close to y, then y is epsilon-close to x.

b) Let epsilon, gamma >0. If x is epsilon-close to y, and y is gamma-close to z, then x and z are (epsilon + gamma)-close.

c) Let epsilon>0. If y and z are both epsilon close to x, and w is between y and z (i.e.. y<=w <=z or z <= w <=y), then w is also epsilon-close to x.

d) Let epsilon >0. If x and y are epsilon-close, and z is non-zero, then xz and yz are epsilon absolute value of z-close

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#### Solution Preview

let x,y,z,w be rational numbers

a) Let epsilon>0. If x is epsilon-close to y, then y is epsilon-close to x.

Given that x is epsilon-close to y. This implies . We know that for any rational number a, hence which implies y is epsilon-close to x.

b) Let epsilon, gamma >0. If x is ...

#### Solution Summary

The expert examines proofs about integers and rationals.