How can I prove this statement by induction? I would like to prove that
$2^0 + 2^{-1} + 2^{-2} +...+ 2^{-n} \leq 2$
by induction.

I got the induction basis as $2^0 \leq 2$, but I couldn't find a induction step that could help me to prove that.

Any suggestions?
Thank you
This is one of those cases where in order for induction to work, you actually need to prove something stronger. Try replacing $\leq 2$ with $=2-2^{-n}$.
$2^0 + 2^{-1} + 2^{-2} +... + 2^{-n} &lt;= 2$

For the first case note that $2^0 \leq 2$ is true and also $0 \leq 2- 2^0$. In fact $2- 2^0 = 2^0$

Next notice that $2^k - 2^{k-1} = 2^{k-1}$

You can prove by induction that $2-(2^0 + 2^{-1} + 2^{-2} +...+ 2^{-n}) = 2^{-n}$