Uniqueness of Limit of Function in Metric Space

Theorem

The definition of a limit of a function in a metric space gives a uniquely defined limit when it exists.

Proof

Consider a metric space X with a∈X, in which the function f is such that limx→af(x)=L1 and limx→af(x)=L2.

We then assume, by way of contradiction, that L1≠L2, which implies that d(L1,L2)>0 from the properties of the metric. As such, let ϵ=d(L1,L2)2. From the definition of the limit, there exists a δ1 such that

0<d(x,a)<δ1⟹d(f(x),L1)<ϵ=d(L1,L2)2

and a δ2 such that

0<d(x,a)<δ2⟹d(f(x),L2)<ϵ=d(L1,L2)2.

However then for δ=min{δ1,δ2} we have by the triangle inequality that

0<d(x,a)<δ⟹d(L1,L2)≤d(f(x),L1)+d(f(x),L2)<d(L1,L2)

a contradiction.