Using two inequality for bounding?

Question

When we know

• $a \leq b - c$
• $c\geq d$,

then $a \leq b - d$ holds?

If so, why does it?

Answer 1

Hint:

$c\ge d \iff c-d\ge 0 \iff 0\le c-d$.

Add that last inequality to $a\le b-c$.

Answer 2

I think a weighing balance is a good analogy for this sort of thing.

Imagine keeping two weights on two pans of the balance. Let one of them be $a$ and the other be $b-c$. Now this balance is either equal or tilted more towards $b-c$. This is the first result we have.

Now if we change the weights to $c$ and $d$, the balance tilts towards $c$. This is our second result.

Now imagine putting 2 weights $b$ in two pans. The balance would be in equilibrium. Now change it to $b-c$ and $b-d$. This is the same as taking away weights $c$ and $d$ from the two pans. Now a greater weight removed will lighten the pan more so this means $b-d$ will be greater than or equal to $b-c$.

Since $b-c$ was itself greater than or equal to $a$ so $b-d$ will be too.