You've seen in the 10th Grade lesson about the sequences what an increasing sequence is, a decreasing sequence, monotonic, bounded above, below, and both.
Now, let's see what a convergent sequence is and what are adjacent sequences. |

Increasing majored sequence |
Adjacent sequences |

Also, note that for two sequences defined this way, you can determine an approximate value of the terms of the sequence and make conjectures about the convergence of the sequence, thanks to a drawing. Then let's draw in an orthornormal frame the graph of f, and let's place on the X-axis the first term . We have then using the graph of f, let's place on the Y-axis the term . Now let's draw the line of equation equation y=x. Come back, from on this line, and go down on the X-axis, so that you put back on the X-axis. Now, using the graph of f, place on the Y-axis and put back its value on the X-axis thanks to the line y=x. This way you can place as many terms of the sequence on the X-axis as you want and guess the limit of the sequence.

1. Suppose ="the property you want to prove", for example here you'll suppose

2. Show that is true. Generally it's rather simple. Here is true because and 0 is a multiple of 3.

3. Show that for every number n, if is true, then is also true. It's the most difficult step. To write the solution, you will for example say : "n is a natural number. Suppose that is true.". We want to show that is also true, that means that is a multiple of 3.

is a multiple of 3.

is a multiple of 3 because is true. The sum of two multiples of 3 is a multiple of 3, hence is a multiple of 3, thus is a multiple of 3, and therefore is true.

4. Then conclude. Since is true, and since for every n, , we have , then is true, then is true... and therefore is true for every n. To say it, just write : "using the induction principle, is true for every n".