Prove that a b &lt; a + c b + d &lt; c d I'm reading a introductory book on mathematical proofs and I am stuck on a question.Let a , b , c , d be positive real numbers, prove that if a b &lt; c d , then a b &lt; a + c b + d &lt; c d

Question

Prove that        a    b    &amp;lt;            a      +      c              b      +      d        &amp;lt;      c    d  I&#039;m reading a introductory book on mathematical proofs and I am stuck on a question.Let   a  ,  b  ,  c  ,  d be positive real numbers, prove that if       a    b    &amp;lt;      c    d  , then       a    b    &amp;lt;            a      +      c              b      +      d        &amp;lt;      c    d

kpgt1z · Accepted Answer

HINT: Note that                                          a            +            c                                b            +            d                          −                  a          b                                    =                              a            +            c                                b            +            d                          ⋅                  b          b                −                  a          b                ⋅                              b            +            d                                b            +            d                                                            =                              b            (            a            +            c            )            −            a            (            b            +            d            )                                b            (            b            +            d            )                                                            =                              a            b            +            b            c            −            a            b            −            a            d                                b            (            b            +            d            )                                                            =                              b            c            −            a            d                                b            (            b            +            d            )                                  .            You’d like to show that this difference is positive, so you’d like to know that   b  c  &amp;gt;  a  d. Use the fact that       a    b    &amp;lt;      c    d   to show this.The other inequality can be handled similarly.

arridsd9 · Accepted Answer

a    b    =            a      (      1      +              d        b            )              b      (      1      +              d        b            )        =            a      +      d      (              a        b            )              b      +      d        &amp;lt;            a      +      d      (              c        d            )              b      +      d        =            a      +      c              b      +      d      and similarly for the other inequality.