Basic Solow Growth Model: Stability Proof
For completeness, let me illustrate this in the continuous time framework. The Solow equation, in the simplest of cases, is

$\dot{k} = s f(k) - \delta k = \phi(k)$

Then we have 

$\frac{\partial \phi}{\partial k} = s f'(k) - \delta = \frac{sf'(k)k - \delta k }{k}$.

In steady state (i.e., $\dot{k} = \phi(k^{\ast}) = 0$), we have $\delta k = s f(k)$, hence

$\left. \frac{\partial \phi}{\partial k} \right|_{k=k^{\ast}} = \frac{s f' k - s f(k)}{k} = \frac{s}{k} f(k) \left[ \frac{k \cdot f'}{f(k)} - 1 \right]$

Since both $\frac{s}{k}$ and $f(k)$ are always positive, the sign of that expression hinges on the term in brackets. But remember that $\frac{k \cdot f'}{f(k)}$ is the output elasticity of capital, which is smaller than 1 if $f$ is concave in $k$, which means $\frac{\partial \phi}{\partial k}$ at $k^{\ast}$ is negative. Therefore by theorem 3.2.1 in Zhang (2005), the steady state in the Solow model is stable.