Prove that `ln (1+1/x) gt (1)/(1+x), x gt 0`. Hence, show that the function `f(x)=(1+1/x)^(x)` strictly increases in `(0, oo)`.

Explanation

To prove that \( \ln\left(1 + \frac{1}{x}\right) > \frac{1}{1+x} \) for \( x > 0 \), we will define a function and analyze its behavior.

### Step 1: Define the function
Let
\[ g(x) = \ln\left(1 + \frac{1}{x}\right) - \frac{1}{1+x} \]
We want to show that \( g(x) > 0 \) for \( x > 0 \).
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