$x_1=1,x_n=x_{n+1}+\ln (1+x_{n+1})$, prove $x_n\leq\frac{1}{2^{n-2}}$. $x_1=1,x_n=x_{n+1}+\ln (1+x_{n+1})$, prove


  
    $x_n\leq\frac{1}{2^{n-2}}$.
  


I proved $0&lt;x_{n+1}&lt;x_n$ by contradiction, and I also get $x_n\geq\frac{1}{2^{n-1}}$ by induction. I tried $\ln(1+x)\geq x-\frac{1}{2}x^2$, but it did not work. Could anyone help me? Thanks!

I tried a new way

$\forall \epsilon&gt;0, \ln(1+x)&gt;(1-\epsilon)x$, then $x_n&gt;(2-\epsilon)x_{n+1}$, thus $x_{n+1}&lt;\frac{1}{(2-\epsilon)2^{n-2}}\to \frac{1}{2^{n-1}}$.

Is it right?
the material you edited in is not enough. We do get an intermediate detail,
$$ 0 &lt; x_{n+1} \; &lt; \; \frac{1}{1 + \sqrt{1 - \frac{x_n}{2} \; }} $$ 
which does convince me that, for example,
$$  2^n x_n $$
is bounded and has a limit, let us call the limit $C.$ The bad news is that they want $C &lt; 4$ and possibly a little more than that information. More work is required.   If $2^n x_n$ decreases with $n$ that would be enough. If, instead, $2^n x_n$ increases with $n$ as well as the limit $C \leq 4,$ that would do it. 

ADDED: I did an experiment just using the rule 
$$  x_{n+1} \; = \; \frac{1}{1 + \sqrt{1 - \frac{x_n}{2} \; }} $$
which suggests that the original sequence has $2^n x_n$ increasing with limit below $3$

Mon May  7 11:16:13 PDT 2018
2   x_n :  0.585786437626905       2^n x_n :  2.34314575050762
3   x_n :  0.3182071694925709       2^n x_n :  2.545657355940568
4   x_n :  0.1659919135906575       2^n x_n :  2.655870617450521
5   x_n :  0.08479343860285271       2^n x_n :  2.713390035291287
6   x_n :  0.04285587582459973       2^n x_n :  2.742776052774383
7   x_n :  0.02154397361204903       2^n x_n :  2.757628622342276
8   x_n :  0.01080115303273365       2^n x_n :  2.765095176379814
9   x_n :  0.005407887829059747       2^n x_n :  2.768838568478591
10   x_n :  0.002705774218059653       2^n x_n :  2.770712799293084
11   x_n :  0.001353344994698495       2^n x_n :  2.771650549142518
12   x_n :  0.0006767870075126321       2^n x_n :  2.772119582771741
13   x_n :  0.0003384221361418738       2^n x_n :  2.77235413927423
14   x_n :  0.0001692182267730049       2^n x_n :  2.772471427448913
15   x_n :  8.46109031377349e-05       2^n x_n :  2.772530074017297
16   x_n :  4.230589901614034e-05       2^n x_n :  2.772559397921774
17   x_n :  2.115306137107151e-05       2^n x_n :  2.772574060029085
18   x_n :  1.057655865143398e-05       2^n x_n :  2.77258139112151
19   x_n :  5.288286317210035e-06       2^n x_n :  2.772585056677415
20   x_n :  2.644144906480589e-06       2^n x_n :  2.77258688945779
21   x_n :  1.322072890209476e-06       2^n x_n :  2.772587805848584
22   x_n :  6.610365543470697e-07       2^n x_n :  2.772588264044132
23   x_n :  3.305183044841222e-07       2^n x_n :  2.772588493141944
24   x_n :  1.652591590697085e-07       2^n x_n :  2.772588607690859
25   x_n :  8.262958124176617e-08       2^n x_n :  2.772588664965319
26   x_n :  4.131479104761107e-08       2^n x_n :  2.772588693602549
27   x_n :  2.065739563048754e-08       2^n x_n :  2.772588707921165
28   x_n :  1.032869784191427e-08       2^n x_n :  2.772588715080472
29   x_n :  5.164348927624759e-09       2^n x_n :  2.772588718660126
30   x_n :  2.582174465479286e-09       2^n x_n :  2.772588720449953
31   x_n :  1.291087233156369e-09       2^n x_n :  2.772588721344867
32   x_n :  6.455436166823664e-10       2^n x_n :  2.772588721792324
33   x_n :  3.227718083672286e-10       2^n x_n :  2.772588722016052
34   x_n :  1.613859041901257e-10       2^n x_n :  2.772588722127916
35   x_n :  8.069295209669067e-11       2^n x_n :  2.772588722183849
36   x_n :  4.03464760487523e-11       2^n x_n :  2.772588722211815
37   x_n :  2.017323802447789e-11       2^n x_n :  2.772588722225798
38   x_n :  1.008661901226438e-11       2^n x_n :  2.772588722232789
39   x_n :  5.043309506138548e-12       2^n x_n :  2.772588722236285
40   x_n :  2.521654753070864e-12       2^n x_n :  2.772588722238033
41   x_n :  1.260827376535829e-12       2^n x_n :  2.772588722238907
42   x_n :  6.30413688268014e-13       2^n x_n :  2.772588722239344
43   x_n :  3.152068441340318e-13       2^n x_n :  2.772588722239562
44   x_n :  1.576034220670221e-13       2^n x_n :  2.772588722239671
45   x_n :  7.880171103351262e-14       2^n x_n :  2.772588722239726
46   x_n :  3.94008555167567e-14       2^n x_n :  2.772588722239754
47   x_n :  1.970042775837845e-14       2^n x_n :  2.772588722239767
48   x_n :  9.850213879189249e-15       2^n x_n :  2.772588722239774
49   x_n :  4.925106939594631e-15       2^n x_n :  2.772588722239778
50   x_n :  2.462553469797317e-15       2^n x_n :  2.77258872223978
51   x_n :  1.231276734898659e-15       2^n x_n :  2.772588722239781
52   x_n :  6.156383674493296e-16       2^n x_n :  2.772588722239781
53   x_n :  3.078191837246648e-16       2^n x_n :  2.772588722239782
54   x_n :  1.539095918623324e-16       2^n x_n :  2.772588722239782
55   x_n :  7.695479593116621e-17       2^n x_n :  2.772588722239782
56   x_n :  3.84773979655831e-17       2^n x_n :  2.772588722239782
57   x_n :  1.923869898279155e-17       2^n x_n :  2.772588722239782
58   x_n :  9.619349491395776e-18       2^n x_n :  2.772588722239782
59   x_n :  4.809674745697888e-18       2^n x_n :  2.772588722239782
60   x_n :  2.404837372848944e-18       2^n x_n :  2.772588722239782
61   x_n :  1.202418686424472e-18       2^n x_n :  2.772588722239782
62   x_n :  6.01209343212236e-19       2^n x_n :  2.772588722239782
63   x_n :  3.00604671606118e-19       2^n x_n :  2.772588722239782
64   x_n :  1.50302335803059e-19       2^n x_n :  2.772588722239782
65   x_n :  7.51511679015295e-20       2^n x_n :  2.772588722239782
66   x_n :  3.757558395076475e-20       2^n x_n :  2.772588722239782
67   x_n :  1.878779197538238e-20       2^n x_n :  2.772588722239782
68   x_n :  9.393895987691188e-21       2^n x_n :  2.772588722239782
69   x_n :  4.696947993845594e-21       2^n x_n :  2.772588722239782
70   x_n :  2.348473996922797e-21       2^n x_n :  2.772588722239782
71   x_n :  1.174236998461398e-21       2^n x_n :  2.772588722239782
72   x_n :  5.871184992306992e-22       2^n x_n :  2.772588722239782
73   x_n :  2.935592496153496e-22       2^n x_n :  2.772588722239782
74   x_n :  1.467796248076748e-22       2^n x_n :  2.772588722239782
75   x_n :  7.33898124038374e-23       2^n x_n :  2.772588722239782
76   x_n :  3.66949062019187e-23       2^n x_n :  2.772588722239782
77   x_n :  1.834745310095935e-23       2^n x_n :  2.772588722239782
78   x_n :  9.173726550479675e-24       2^n x_n :  2.772588722239782
79   x_n :  4.586863275239838e-24       2^n x_n :  2.772588722239782
80   x_n :  2.293431637619919e-24       2^n x_n :  2.772588722239782
81   x_n :  1.146715818809959e-24       2^n x_n :  2.772588722239782
82   x_n :  5.733579094049797e-25       2^n x_n :  2.772588722239782
83   x_n :  2.866789547024899e-25       2^n x_n :  2.772588722239782
84   x_n :  1.433394773512449e-25       2^n x_n :  2.772588722239782
85   x_n :  7.166973867562246e-26       2^n x_n :  2.772588722239782
86   x_n :  3.583486933781123e-26       2^n x_n :  2.772588722239782
87   x_n :  1.791743466890562e-26       2^n x_n :  2.772588722239782
88   x_n :  8.958717334452808e-27       2^n x_n :  2.772588722239782
89   x_n :  4.479358667226404e-27       2^n x_n :  2.772588722239782
90   x_n :  2.239679333613202e-27       2^n x_n :  2.772588722239782
91   x_n :  1.119839666806601e-27       2^n x_n :  2.772588722239782
92   x_n :  5.599198334033005e-28       2^n x_n :  2.772588722239782
93   x_n :  2.799599167016502e-28       2^n x_n :  2.772588722239782
94   x_n :  1.399799583508251e-28       2^n x_n :  2.772588722239782
95   x_n :  6.998997917541256e-29       2^n x_n :  2.772588722239782
96   x_n :  3.499498958770628e-29       2^n x_n :  2.772588722239782
97   x_n :  1.749749479385314e-29       2^n x_n :  2.772588722239782
98   x_n :  8.74874739692657e-30       2^n x_n :  2.772588722239782
99   x_n :  4.374373698463285e-30       2^n x_n :  2.772588722239782
100   x_n :  2.187186849231643e-30       2^n x_n :  2.772588722239782
Mon May  7 11:16:13 PDT 2018
well, I think I misunderstand this question, sorry.

as you said

$x_{n+1}\ge ln(1+x_{n+1})\geq x-\frac{1}{2}x^2$

we can get $x_n\in(\frac{1}{2^{n-1}}, 2-\sqrt{4-2x_{n-1}})$

Thus we need to show $2-\sqrt{4-2x_{n-1}}\le \frac{1}{2^{n-2}}$

in fact, the above $2-\sqrt{4-2x_{n-1}}$ is determined by $ 2-\sqrt{2\sqrt{2\sqrt{2...}}}$ which you can check with $\frac{1}{2^{n-2}} $