If s1=√2,
and sn+1=√2+√sn
Prove that sn is converging, and that sn<2 for all n=1,2…
My attempt to use mathematical induction: if sn<2 then I use
sn+1=√2+√sn, to prove sn+1<√2+2=2.
I guess sn is an increasing sequence, I can use the sn is bounded and increasing, and then prove that sn converges.
Answer
To give a name to your idea, you want to use the monotone convergence theorem; a bounded and increasing sequence converges to the supremum of that set, indeed a standard approach would be to use induction.
Lemma: sn is bounded
Base case:
n=1
s1=√2<2
Suppose now for arbitrary k
sk<2
Now observe as the inductive step:
sk+1=√2+√sk<√2+2=2
By the principle of mathematical induction we have found that 2 is a bound for all n∈N.
Lemma: sn is increasing
Base case:
s2−s1>0
√2+√√2−√2>√2+0−√2=0
Now suppose for arbitrary k we have:
sk−sk−1>0⟹sk>sk−1⟹√sk>√sk−1
We now make our inductive step:
sk+1−sk=√2+√sk−√2+√sk−1>√2+√sk−1−√2+√sk−1=0
Hence we have an increasing sequence for all n∈N this completes the proof and our sequence is indeed convergent ◻.
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