Proving the convergence of a sequence

[transgalactic]

$$2 ,2+\frac{1}{2},2+\frac{1}{2+\frac{1}{2}}$$
etc..
(the sequence consists only from positive number so the sum is not negative)
in order to prove that its convergent i need to prove monotonicity and boundedness

monotonicity:(by induction)

$$a_1=2$$
$$a_2=2.5$$
so i guess its increasing
suppose n=k is true:
$$a_{k-1}<a_k$$
prove n=k+1 ($$a_{k}<a_{k+1}$$)
$$a_k>a_{k-1}\\$$
$$\frac{1}{a_k}<\frac{1}{a_{k-1}}\\$$
$$2+\frac{1}{a_k}<2+\frac{1}{a_{k-1}}\\$$
$$a_{k+1}<a_k$$

i proved the opposite :)
so this is weird.

the answer in the book tells me to split the sequence into odd /even sub sequences
the one is ascending and the other its descending.

i can't see how many sub sequences i need to split it to
maybe its 5 or 10
what is the general way of solving it.
and how you explained that i proved the opposite

 

[mighty2000]

.To prove monotonicity, you need to show that the sequence is either increasing or decreasing. In your attempt, you assumed that the sequence is increasing and tried to prove it, but you ended up proving that it is decreasing. This is not correct, as you cannot assume the property you are trying to prove.

To prove monotonicity, you can split the sequence into two subsequences, one consisting of all the even terms and the other consisting of all the odd terms. Let's call these subsequences a_n and b_n, respectively.

For the even terms (a_n):
a_1=2
a_2=2+\frac{1}{2}=2.5
a_3=2+\frac{1}{2+\frac{1}{2}}=2.6
a_4=2+\frac{1}{2+\frac{1}{2+\frac{1}{2}}}=2.625
a_5=2+\frac{1}{2+\frac{1}{2+\frac{1}{2+\frac{1}{2}}}}=2.629
...

We can see that the even terms are increasing, as each term is greater than the previous one. This can be proved by induction as you attempted in your answer.

For the odd terms (b_n):
b_1=2+\frac{1}{2}=2.5
b_2=2+\frac{1}{2+\frac{1}{2}}=2.6
b_3=2+\frac{1}{2+\frac{1}{2+\frac{1}{2}}}=2.625
b_4=2+\frac{1}{2+\frac{1}{2+\frac{1}{2+\frac{1}{2}}}}=2.629
...

We can see that the odd terms are decreasing, as each term is less than the previous one. This can also be proved by induction.

Since both subsequences are monotonic (one increasing and one decreasing), the original sequence must also be monotonic. This shows that the sequence is bounded.

To show that the sequence is bounded, we can use the fact that the even terms are increasing and the odd terms are decreasing. This means that the even terms are always greater than or equal to the odd terms. So, we can say that:

a_n≥b_n for all n

Since the even terms