Fraction increases on adding 1?

Number Theory Level pending

Shyam made the following observations:

\[\dfrac{3+1}{4+1} > \dfrac 34 \\ \dfrac{1+1}{2+1} > \dfrac 12 .\]

From these observations, he concluded that for every​ natural number \(a\) and \(b\), the following inequality always holds.

\[\dfrac{a+1}{b+1} > \dfrac ab\]

Is he correct?

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