Two inequalities involving real positive numbers Let $a, b, c, d$ be some real positive numbers. I would like to understand if these inequalities are true:
$1) \max(a, b)\leq\max(a+c, b+d);$
$2) \max(a, b) +\max(c, d)\geq\max(a+c, b+d).$
About me the first inequality is trivially true because we deal with real positive numbers. Moreover, also the second inequality is true (about me). I proceed in this way:

*

*if $\max(a,b)=a$ and $\max(c, d)=c$, thus $a+c=a+c$;


*if $\max(a,b)=a$ and $\max(c, d)=d$, thus $a+d> a+c$ and $a+d> b+d$ and in particular it is greater than the maximum;


*if $\max(a,b)=b$ and $\max(c, d)=c$, thus $b+c>a+c$ and $b+c> b+d$ and in particular it is greater than the maximum;


*if $\max(a,b)=b$ and $\max(c, d)=d$, thus $b+d = b+d$.
Could anyone please tell me if my reasonings are valid and if the two inequalities are true?
Thank you in advance!
 A: Yes, the inequalities are true and your reasonings are correct. From the very casework that you've done, you can see that
$$\max(a, b) + \max(c, d) = \max(a + c, a + d, b + c, b + d)$$
and hopefully from here it should be clear why the second inequality holds.
Note that the second inequality holds for any real numbers $a, b, c, d$.
A: Both inequalities are true, but your reasoning for the second one is not completely correct. For example:


*

*if $\max(a,b)=a$ and $\max(c, d)=d$, thus $a+d> a+c$ and $a+d> b+d$ and in particular it is greater than the maximum;


You can only conclude that $a+d \ge a+c$ and $a+d \ge b+d$ in that case, however that is sufficient to get the desired estimate. The same applies to the case


*

*if $\max(a,b)=b$ and $\max(c, d)=c$, thus $b+c>a+c$ and $b+c> b+d$ and in particular it is greater than the maximum;



A perhaps simpler argument is that
$$
 a + c \le \max(a, b) +\max(c, d)
$$
and
$$
 b + d \le \max(a, b) +\max(c, d)
$$
and therefore
$$
\max(a+c, b+d) \le \max(a, b) +\max(c, d) \, .
$$
