Mathematical induction is done by proving that the first statement in the infinite sequence of statements is true, and then proving that if any one statement in the infinite sequence of statements is true, then so is the next one.

Similarly, I can prove a proposition by proving a proposition is true when x=x0 and then proving that if any x can make the proposition true, then so is x+Δx(Δx>0 andΔx→0).

e.g.:

prove e^(i m)=cos(m)+i sin(m)(Euler’s formula)

First, when m=0,obviously,it's true.

Second, assume it's true when m=x,then when m= x+Δx(Δx>0 andΔx→0),

e^i(x+∆x) ={d/dx[e^(ix)]}∆x+e^(ix)=e^(ix)+i e^(ix) ∆x=cos(x)+i sin(x)+i[cos(x)+i sin(x) ]∆x=[cos(x)-sin(x)∆x]+i[sin(x)+cos(x)∆x]

∵∆x→0,∴cos(∆x)=1,sin(∆x)=∆x;

∴cos(x+∆x)=cos(x)cos(∆x)-sin(x)sin(∆x)=cos(x)-sin(x)∆x;

sin(x+∆x)=sin(x)cos(∆x)+cos(x)sin(∆x)=sin(x)+cos(x)∆x;

∴[cos(x)-sin(x)∆x]+i[sin(x)+cos(x)∆x]=cos(x+∆x)+i sin(x+∆x)

Therefore,if the formula is true for m=x,the formula is also true for m= x+Δx.

It's easy to prove if the formula is true for m=x,the formula is also true for m=x-Δx

So the formula is true for any m.

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Note:

1.I'm not a native English speaker. So there maybe many errors on language.If you don't understand what I'm saying,please tell me and I would correct the mistakes.I would be glad if you point our my errors on language.The reason why I come to Physics Forums is that forums in my country are either unpopular or aimed to college entrance exam,which makes hundreds of thousands of students each province work hard to get a higher score in order to be admitted into a good university,rather than intend for math and and physics fans.

2.If my idea is wrong,please tell me where the error is.