<s>[INST] Inequality Induction Proof, how should I proceed? I have been trying to prove inequalities using induction to no avail.
For example,
Prove the following:
$\frac{1}{2^{1}}+\frac{2}{2^{2}}+\frac{3}{2^{3}}+...+\frac{n}{2^{n}}<2$
Base Case:
$\frac{1}{2^{1}} = 2$
$\frac{1}{2}<2\;$  Which is true.
We assume, for n=k, that
$\frac{1}{2^{1}}+\frac{2}{2^{2}}+\frac{3}{2^{3}}+...+\frac{k}{2^{k}}<2\;\;\;$ is true.
As induction step with n=k+1, we need to prove that
$\frac{1}{2^{1}}+\frac{2}{2^{2}}+\frac{3}{2^{3}}+...+\frac{k+1}{2^{k+1}}<2\;\;\;$
How should I proceed from here?
 [/INST] You may first prove the following equality:

Prove that $\frac 1 {2^1} + \frac 2 {2^2} + \dots + \frac k {2^k} = 2 - \frac {k + 2} {2^k}$.

This can be proved by induction on $k$ (easy exercise).
Once it is proved, the original inequality follows.
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