Is it true that ||z| - |w|| \leq |z + w| ?

  • Thread starter smoothman
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  • #1
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is it true that ||z| - |w|| \leq |z + w| ??

is it true that

[itex] ||z| - |w|| \leq |z + w| [/itex]??

if so what is the proof?

here is my workin so far.. please verify it thanks.

We know [itex] |z+w| \leq |z|+|w| [/itex]
let c = z - w, so [itex] |c+w| \leq |c|+|w| [/itex]

Now z = c + w,
so [itex] |z| \leq |c|+|w| [/itex]
[itex] |z| \leq |z-w| + |w| [/itex]
[itex] |z-w| \geq |z| -|w| [/itex]

Now let d = - w,
so [itex] |z - d| \geq |z| - |d| [/itex]
subbing in -w for d we get, [itex] |z + w| \geq |z| - |d| [/itex]
subbing in |w| for |d| since they are equal, we get [itex] |z + w| \geq ||z| - |w|| [/itex]
(i added an extra modulus bracket outside the right hand side at the end of the equation).

End of proof.

Is this correct please? please guide me if i am wrong?
 
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Answers and Replies

  • #2
486
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|z| =|z+w-w|<= |z+w|+|w|

go from here
 

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