Q1- Define dA : X → R by dA(x) = inf{d(x,y) : y ∈ A}.
Prove that dA is bounded and uniformly continuous. Moreover, show that for all x,y ∈ X, |dA(x) − dA(y)| ≤ d(x, y).
The function dA measures how close is the point x from A. Now, Let Aε ={x∈X :dA(x)<ε}. We refer to Aε as the ε-neighborhood of A. For every finite partition γ of X and every ε > 0, let γ(ε) = ? Aε × Aε. A∈γ   Q2- Let (X, d) be a metric space. Define dˆ : X × X → R, by: ˆ
d(x, y) = min{1, d(x, y)}. (a) Prove that dˆ is a bounded metric on X. (b) Use part (a) to prove that for ε > 0 there exists a bounded metric dˆ on X such that for all ˆ
x,y∈X we have d(x,y)<1⇒d(x,y)<ε.

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Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.