Proof by induction, $1/2 + ... + n/2^n < 2$ So I'm having trouble with proving this homework question by induction. 
$$
\frac{1}{2^1} + \frac{2}{2^2} + ... +\frac{n-1}{2^{n-1}} + \frac{n}{2^n} <2
$$
I know how to prove that the series converges to 2 (using things like the ratio method), but actually using induction is where I get confused.
Base case is easy, n=1.
$$
\frac{1}{2^1}<2
$$
Induction case we assume that
$$
\frac{1}{2^1} + \frac{2}{2^2} + ... +\frac{k-1}{2^{k-1}} + \frac{k}{2^k} <2
$$
Then we get to fun old induction. How do I show that
$$
\frac{1}{2^1} + \frac{2}{2^2} + ... +\frac{k-1}{2^{k-1}} + \frac{k}{2^k} + \frac{k+1}{2^{k+1}} <2 ?
$$

In that case a possible trick to use induction is to prove the stronger condition
$$\frac{1}{2} + \frac{2}{2^2} +  \ldots +\frac{n-1}{2^{n-1}} + \frac{n}{2^{n}} <2-\frac{n+1}{2^{n-1}}<2$$
and the induction step becomes
$$\frac{1}{2} + \frac{2}{2^2} + \ldots + \frac{n}{2^n} + \frac{n+1}{2^{n+1}}\stackrel{Ind. Hyp.}<2-\frac{n+1}{2^{n-1}}+ \frac{n+1}{2^{n+1}}\stackrel{?}<2-\frac{n+2}{2^{n}}$$
and the last inequality holds indeed
$$2-\frac{n+1}{2^{n-1}}+ \frac{n+1}{2^{n+1}}\stackrel{?}<2-\frac{n+2}{2^{n}}$$
$$\frac{n+1}{2^{n-1}}- \frac{n+1}{2^{n+1}}\stackrel{?}>\frac{n+2}{2^{n}}$$
$$4(n+1)-(n+1)\stackrel{?}>
2(n+2)$$
$$3n+3\stackrel{?}>2n+4$$
wich is true for $n>1$.
