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Earlier today I was trying to prove that if a limit of a certain function exists, then it's unique:

limf(x)=a [itex]\wedge[/itex] limf(x)=b (as x→x_{0}) then a=b

I began to use the sum of limits like so:

limf(x)+limf(x)=a+a → lim2f(x)=2a (as x→x_{0})

And the same thing for limf(x)=b results in lim2f(x)=2b.

Now, I thought that if limf(x)=a [itex]\wedge[/itex] limf(x)=b, then:

lim2f(x)=a+b (as x→x_{0})

I concluded that a+b=2b [itex]\vee[/itex] a+b=2a, which gave me a=b on both equations.

Is this proof acceptable or do I have to prove it by the definition of limit?

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# Is this proof (about limits) acceptable?

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