Why is convexity of $S$ needed in these questions?? (Bachman &Narici, Q-17,18)

Question

Let $X$ be a Hilbert Space and let $\{S\}$ be a Convex set in $X$. Let $d=\inf_{x \in S}\|x\|$ . Prove that, if $\{x_n\}$ is a sequence of elements in $S$ such that $\lim_n \|x_n\|=d$, then $\{x_n\}$ is a Cauchy sequence.

This is a problem from the book "Functional Analysis" by Bachman and Narici ( Page-161, Q.17). I don't see why convexity of $S$ is needed and again any Normed space would have done. I guessed that all these are needed for the next question (Q.18) which is a continuation of this one.

The next Question states that :

Show that any closed convex subset $S$ of a Hilbert Space $X$ contains a unique element of the minimal form.

Suppose that $d=\inf_{x \in S}\|x\|$. Then $d+\dfrac{1}{n}$ is no longer an infimum and hence there is a $x_n \in S$ such that $\|x_n\| < d+\dfrac{1}{n}$. Then $d \le \|x_n\|=y_n < d+\dfrac{1}{n}$ and letting $n \to \infty$ we have $\lim\|x_n\|=d$

By Previous Exercise $x_n$ is a cauchy sequence in $S$ and since $S$ is closed and hence Complete implying $x_n \to x \in S$ and $\|x\|=d$.

I don't see why Convexity of $S$ is needed and a Banach Space would have done in place of Hilbert Space.

Thanks for the help!!

I found my mistake in the proof of the first part, I am given that $ ||x_n|| \to d$ where as I made use of $x_n$ going to $d$. — tattwamasi amrutam, Aug 23 '15 at 11:52
For the 1st Q, with infinite-dimensional $X$, convexity is needed.For example let $S= { v \in X : 1 \le ||v|| \le 2 }$ and let ${x_n : n \in N }$ be a Hilbert -space basis for $X$ with $||v_n||=1$ for each $n \in N$. — DanielWainfleet, Aug 23 '15 at 14:59
If the first Q holds in any Banach space then so does the second,as your proof will apply verbatim. If not then the inner product properties will be needed in the first one.I suggest seeing whether you can do the 1st using inner products in some way, and then see if you can do without it. — DanielWainfleet, Aug 23 '15 at 15:05
I am trying to use convexity and inner product ness to solve. — tattwamasi amrutam, Aug 23 '15 at 15:30
Can someone give any hint to try and show that the sequence is cauchy?? — tattwamasi amrutam, Aug 24 '15 at 07:40

Why is convexity of $S$ needed in these questions?? (Bachman &Narici, Q-17,18)

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