trouble solving this sequence problem I'm having some trouble solving this problem about sequences:

$$a(n): a(1) = 2;$$  $$   a(n+1) = (a(n) + 1)/2, n\in N$$

1)Prove that this sequence is monotonically decreasing

2)Prove that$$ 1 &lt; a(n) ≤ 2, \forall n\in N$$

3)Prove that this sequence is convergent and calculate its limit

The first one, after calculating some terms, I saw that it was decreasing to 1, but how do I prove it theoretically?
I should start with 2) using induction.

2) If $1&lt;a_n\leq 2$ then: $$a_{n+1}=\frac{a_n+1}{2}&gt;\frac{1+1}{2}=1$$ and $$a_{n+1}=\frac{a_n+1}{2}\leq\frac{2+1}{2}\leq 2$$

1) From $a_n&gt;1$ it follows directly that: $$a_{n+1}=\frac{a_n+1}{2}&lt;a_n$$

A bounded monotone sequence is convergent and we can find its limit $a$ on base of the relation $a_{n+1}=\dfrac{a_n+1}{2}$. Both sides converges and this leads to $a=\dfrac{a+1}{2}$ hence $a=1$
$2[a(n+1)-1]=a(n)-1$
$a(n+1)-1=(1/2)^n$
$a(n+1)=(1/2)^n+1$