Properties of Two Inequalities

Question

If $a < b$ and $c < d$, then $a + c < b + d$ if $a, b, c, d$ are positive integers.

My hunch is that this is true, but I'm having difficulty proving it. Is there a property or definition pertaining to inequalities that exists to support/disprove the above?

Answer 1

$a < b \implies a+c < b+c$

$c < d \implies b+c < b+d$

So $a < b$ and $c<d \implies a+c < b+c < b+d \implies a+c < b+d$

Answer 2

From the axioms of an ordered field, $a<b \implies a+c<b+c$, and $c<d\implies b+c<b+d.$

Therefore, $a+c<b+c<b+d.$