Question Number 164623 by mathocean1 last updated on 19/Jan/22
![Given a, b ∈ R. Show that : [a]+[b]≤[a+b]≤[a]+[b]+1](Q164623.png)
$${Given}\:{a},\:{b}\:\in\:\mathbb{R}. \\ $$$${Show}\:{that}\:: \\ $$$$\left[{a}\right]+\left[{b}\right]\leqslant\left[{a}+{b}\right]\leqslant\left[{a}\right]+\left[{b}\right]+\mathrm{1} \\ $$
Answered by mahdipoor last updated on 19/Jan/22
![[b]≤b<[b]+1⇒a+[b]≤a+b<a+[b]+1⇒ [a+[b]]≤[a+b]≤[a+[b]+1]⇒ [a]+[b]≤[a+b]≤[a]+[b]+1](Q164631.png)
$$\left[{b}\right]\leqslant{b}<\left[{b}\right]+\mathrm{1}\Rightarrow{a}+\left[{b}\right]\leqslant{a}+{b}<{a}+\left[{b}\right]+\mathrm{1}\Rightarrow \\ $$$$\left[{a}+\left[{b}\right]\right]\leqslant\left[{a}+{b}\right]\leqslant\left[{a}+\left[{b}\right]+\mathrm{1}\right]\Rightarrow \\ $$$$\left[{a}\right]+\left[{b}\right]\leqslant\left[{a}+{b}\right]\leqslant\left[{a}\right]+\left[{b}\right]+\mathrm{1} \\ $$
Commented by mathocean1 last updated on 19/Jan/22
$${thanks} \\ $$