proof or disprove

Question

how do I go about proving this statement. I know i can disprove a statement by using its negation but how do I go about solving this statements

∀a, b ∈ R, a ≤ b ⇒ ⌈a⌉ ≤ ⌈b⌉.

∀a, b ∈ R, ⌈a⌉ ≤ ⌈b⌉⇒ a ≤ b

Answer 1

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Ceiling[b]>=b and Ceiling[a]>=a;

Since a<=b, then b>=a , which means b-a >=0

So then

Ceiling[b] - Ceiling[a] >= b - a >= 0

Therefore Ceiling[b]>=Ceiling[a] which proves

the first condition --->

<-----

Suppose a>b.

Then a-b>0

Ceiling[b]>=Ceiling[a] means

Ceiling[b]-Ceiling[a]>=0

So then

Ceiling[a]-Ceiling[b]>=a-b > 0

which means Ceiling[a]>Ceiling[b]

which is a contradiction.

THerefore a<=b <-- if and only if --> Ceiling[a]<=Ceiling[b]