Prove that if $a_1\leq b_1\leq c_1$ and $a_2\leq b_2\leq c_2$, then $a_1+a_2\leq b_1+b_2\leq c_1+c_2$.

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Lesson Parent: 
Rules of Inequalities
Problem: 

Prove that if $a_1\leq b_1\leq c_1$ and $a_2\leq b_2\leq c_2$, then $a_1+a_2\leq b_1+b_2\leq c_1+c_2$.

Answer: 

It is true that if $a_1\leq b_1\leq c_1$ and $a_2\leq b_2\leq c_2$, then $a_1+a_2\leq b_1+b_2\leq c_1+c_2$.

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Solution: 

Let's break it into parts. First, we know that <Sign in to see all the formulas> and <Sign in to see all the formulas>. From this problem we know that

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Second, we know that <Sign in to see all the formulas> and <Sign in to see all the formulas>. From the same problem as above we know that

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The proof is now complete. We know that

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